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Semiparametric Structural Equation Models with Interval-Censored Data

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Title: Semiparametric Structural Equation Models with Interval-Censored Data
Language: English
Authors: Hongxi Li, Shuwei Li, Liuquan Sun, Xinyuan Song (ORCID 0000-0002-4877-3200)
Source: Structural Equation Modeling: A Multidisciplinary Journal. 2025 32(1):14-25.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 12
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Descriptors: Statistical Analysis, Statistical Studies, Structural Equation Models, Intervals, Censorship, Data, Nonparametric Statistics, Maximum Likelihood Statistics, Algorithms, Simulation, Alzheimers Disease, Computation, Mathematical Formulas
DOI: 10.1080/10705511.2024.2368023
ISSN: 1070-5511
1532-8007
Abstract: Structural equation models offer a valuable tool for delineating the complicated interrelationships among multiple variables, including observed and latent variables. Over the last few decades, structural equation models have successfully analyzed complete and right-censored survival data, exemplified by wide applications in psychological, social, or genomic studies. However, the existing methodology for structural equation modeling is not concerned with interval-censored data, a type of coarse survival data arising typically from periodic examinations for the occurrence of asymptomatic disease. The present study aims to fill this gap and provide a flexible semiparametric structural equation modeling framework. A general class of factor-augmented transformation models is proposed to model the interval-censored outcome of interest in the presence of latent risk factors. An expectation-maximization algorithm is subtly designed to conduct the nonparametric maximum likelihood estimation. Furthermore, the asymptotic properties of the proposed estimators are established by leveraging the empirical process theory. The numerical results obtained from extensive simulations and an application to the Alzheimer's disease data set demonstrate the proposed method's empirical performance and practical utility.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1457160
Database: ERIC
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  Value: <anid>AN0182192620;7mz01jan.25;2025Jan15.04:19;v2.2.500</anid> <title id="AN0182192620-1">Semiparametric Structural Equation Models with Interval-Censored Data </title> <p>Structural equation models offer a valuable tool for delineating the complicated interrelationships among multiple variables, including observed and latent variables. Over the last few decades, structural equation models have successfully analyzed complete and right-censored survival data, exemplified by wide applications in psychological, social, or genomic studies. However, the existing methodology for structural equation modeling is not concerned with interval-censored data, a type of coarse survival data arising typically from periodic examinations for the occurrence of asymptomatic disease. The present study aims to fill this gap and provide a flexible semiparametric structural equation modeling framework. A general class of factor-augmented transformation models is proposed to model the interval-censored outcome of interest in the presence of latent risk factors. An expectation–maximization algorithm is subtly designed to conduct the nonparametric maximum likelihood estimation. Furthermore, the asymptotic properties of the proposed estimators are established by leveraging the empirical process theory. The numerical results obtained from extensive simulations and an application to the Alzheimer's disease data set demonstrate the proposed method's empirical performance and practical utility.</p> <p>Keywords: Interval censoring; joint model; nonparametric maximum likelihood estimation; structural equation modeling</p> <hd id="AN0182192620-2">1. Introduction</hd> <p>As many scientific investigations exemplify, gathering a wealth of correlated variables is often possible. Extracting the common information summarized by the correlated variables remains of considerable interest to practitioners (Bollen, [<reflink idref="bib2" id="ref1">2</reflink>]; Dahly et al., [<reflink idref="bib4" id="ref2">4</reflink>]; Naliboff et al., [<reflink idref="bib23" id="ref3">23</reflink>]). Over the past several decades, structural equation modeling (SEM) has been recognized as a potent and valuable tool to delineate the complicated relationships among multiple variables, including latent and observed variables (Fan et al., [<reflink idref="bib8" id="ref4">8</reflink>], [<reflink idref="bib9" id="ref5">9</reflink>]; Lee, [<reflink idref="bib17" id="ref6">17</reflink>]; Liu et al., [<reflink idref="bib20" id="ref7">20</reflink>]; Song & Lee, [<reflink idref="bib26" id="ref8">26</reflink>]; Wong et al., [<reflink idref="bib34" id="ref9">34</reflink>]; Yang et al., [<reflink idref="bib35" id="ref10">35</reflink>]). A typical SEM framework involves a measurement equation and a structural equation. In particular, the interrelationships between the observed and latent variables are delineated via the measurement equation, while the latent variables are linked through the structural equation. Since the seminal work of Bollen ([<reflink idref="bib2" id="ref11">2</reflink>]), SEM methods have been tremendously applied in various fields, including but not limited to psychological studies (Evans et al., [<reflink idref="bib7" id="ref12">7</reflink>]), sociological investigations (Tarka, [<reflink idref="bib31" id="ref13">31</reflink>]), and genomics (Grotzinger et al., [<reflink idref="bib10" id="ref14">10</reflink>]).</p> <p>In survival analysis, SEM is also commonly leveraged to reveal the complex relationships among variables and investigate the effects of latent risk factors on the failure event of interest. Numerous efforts have been devoted to developing SEM methods under various settings (Asparouhov et al., [<reflink idref="bib1" id="ref15">1</reflink>]; Larsen, [<reflink idref="bib15" id="ref16">15</reflink>], [<reflink idref="bib16" id="ref17">16</reflink>]; Stoolmiller & Snyder, [<reflink idref="bib27" id="ref18">27</reflink>], [<reflink idref="bib28" id="ref19">28</reflink>]; Wong et al., [<reflink idref="bib34" id="ref20">34</reflink>]). For example, Larsen ([<reflink idref="bib16" id="ref21">16</reflink>]) used the proportional hazards (PH) model with a single latent variable to capture the association between the survival time and observed variables. Asparouhov et al. ([<reflink idref="bib1" id="ref22">1</reflink>]) formulated a more general relationship among latent and observed variables. Stoolmiller and Snyder ([<reflink idref="bib27" id="ref23">27</reflink>]) explored an SEM framework with a PH submodel for the multivariate failure times. Pan et al. ([<reflink idref="bib24" id="ref24">24</reflink>]) introduced latent variables into additive hazards model to investigate latent risk factors for the hazard of interest. Wong et al. ([<reflink idref="bib34" id="ref25">34</reflink>]) proposed a general semiparametric SEM framework in which a class of transformation models is assumed for the failure time of interest. He et al. ([<reflink idref="bib11" id="ref26">11</reflink>]) considered a joint model to incorporate latent variables into additive mean residual life models. Despite the existing fruitful literature, all the methods above only accommodate data subjected to right censoring.</p> <p>In many practical applications, the failure time of interest frequently suffers from interval censoring rather than simpler right censoring owing to intermittent follow-ups to examine the failure event occurrence (Sun, [<reflink idref="bib30" id="ref27">30</reflink>]). Unlike the right-censored data, which may include some precisely measured failure times, interval-censored data only provide time intervals that bracket failure times instead of exact ones. Compared with right censoring, interval-censored data typically bring additional theoretical and computational challenges. Therefore, extensive efforts have been devoted to their methodology development (Sun, [<reflink idref="bib29" id="ref28">29</reflink>]; Turnbull, [<reflink idref="bib32" id="ref29">32</reflink>]; Wang et al., [<reflink idref="bib33" id="ref30">33</reflink>]; Zeng et al., [<reflink idref="bib38" id="ref31">38</reflink>]). While SEM has been widely investigated in the circumstances of complete and right-censored data, as introduced above, an SEM framework in the context of interval censoring is still unexplored.</p> <p>We propose a general SEM framework to fill this critical gap that can accommodate interval-censored failure time data. Our work is motivated by an integrative analysis from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study. In this study, a primary interest is to investigate the critical risk factors for Alzheimer's disease (AD) progression. The time to AD is an interval-censored outcome due to the use of periodic examinations for the AD occurrence. As commented by Desgranges et al. ([<reflink idref="bib5" id="ref32">5</reflink>]), Drijgers et al. ([<reflink idref="bib6" id="ref33">6</reflink>]), and others, the degree of cognitive impairment may be a potential risk factor influencing the progression of neurodegeneration to dementia. However, the degree of cognitive impairment is a latent variable jointly measured by multiple cognitive test scores. Meanwhile, memory and executive function, which are highly associated with the degree of cognitive impairment, are also latent variables integrated by some measured surrogates as discussed in Park et al. ([<reflink idref="bib25" id="ref34">25</reflink>]) and our Section 6. Therefore, SEM, as a valuable tool to investigate the complex interrelationships among latent and observed variables, is particularly useful in this ADNI data analysis.</p> <p>The proposed SEM framework consists of a class of semiparametric factor-augmented transformation models to examine the effects of latent and observed risk factors on the failure time of interest. The class of transformation models is quite flexible since it comprises the popular PH and proportional odds (PO) models as special cases. In addition, a set of generalized linear models with proper identifiability conditions characterizes latent variables through multiple observed variables. We perform the nonparametric maximum likelihood estimation (NPMLE) and propose a computationally stable and reliable expectation–maximization (EM) algorithm to maximize the intractable likelihood that involves multiple integrations. We use the empirical process techniques to establish the consistency of the resulting estimators and the asymptotic normality of the estimators related to the parametric components. Furthermore, the estimators of the parametric components in the proposed SEM are shown to be asymptotically efficient with the limiting covariance matrix that can be consistently estimated via a profile likelihood approach.</p> <p>The rest of this article is organized as follows. Section 2 describes notations, the proposed structural equation model, the observed data likelihood, and model identifiability conditions. Section 3 presents the proposed NPMLE approach along with the EM algorithm. The asymptotic properties of the resulting estimators are established in Section 4. Section 5 evaluates the empirical performance of the proposed method through simulations, followed by an application to the ADNI study in Section 6. Section 7 concludes the article. Technical details and additional numerical results are provided in the Supplementary Material.</p> <hd id="AN0182192620-3">2. SEM Framework</hd> <p></p> <hd id="AN0182192620-4">2.1. Notation, Model and Likelihood</hd> <p>We propose a semiparametric SEM framework that involves three major components and time-independent covariates. The first component is a structural equation to link latent variables. The second component is the measurement equation to relate latent and observed variables. The third component is a class of semiparametric factor-augmented transformation models to examine the effects of latent factors and observed covariates on the failure time of interest.</p> <p>Suppose a failure time study involving <emph>n</emph> independent subjects. For</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow><mo>,</mo></math> </ephtml> let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> denote a</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi><mo>×</mo><mn>1</mn></mrow></math> </ephtml> vector of latent variables,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> denote a</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo>×</mo><mn>1</mn></mrow></math> </ephtml> vector of observed manifest variables, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> denote an</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>×</mo><mn>1</mn></mrow></math> </ephtml> vector of observed covariates. We first specify the conditional distribution of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>∼</mo><msub><mrow><mi>F</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib1" id="ref35">1</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>F</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><mo>·</mo><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> denotes a <emph>q</emph>-variate normal distribution function and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ν</mi></mrow></math> </ephtml> is a vector containing all unknown parameters in model (<reflink idref="bib1" id="ref36">1</reflink>). For instance, consider a model consisting of two latent variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> and a covariate <emph>Z<subs>i</subs></emph> with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow><mo>.</mo></math> </ephtml> Given <emph>Z<subs>i</subs></emph>,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></math> </ephtml> are assumed to follow the normal distributions</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>ν</mi></mrow><mn>1</mn></msub><msub><mrow><mi>Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msub><mrow><mi>ν</mi></mrow><mn>2</mn></msub><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ν</mi></mrow><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>ν</mi></mrow><mn>4</mn></msub><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> respectively. Thus,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ν</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ν</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>ν</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>ν</mi></mrow><mn>3</mn></msub><mo>,</mo><msub><mrow><mi>ν</mi></mrow><mn>4</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow><mo>,</mo></math> </ephtml> where <emph>ν</emph><subs>1</subs> and <emph>ν</emph><subs>2</subs> are two unknown regression parameters, and <emph>ν</emph><subs>3</subs> and <emph>ν</emph><subs>4</subs> are two unknown variance parameters.</p> <p>Next, given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> the conditional distribution of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is specified as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>∼</mo><msub><mrow><mi>F</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib2" id="ref37">2</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>F</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><mo>·</mo><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> denotes a <emph>p</emph>-variate parametric distribution function indexed by the parameter vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ψ</mi></mrow></math> </ephtml> and determined by the type of each <emph>Y<subs>ij</subs></emph>, the <emph>j</emph>th component of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub></mrow><mo>.</mo></math> </ephtml> Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mi>i</mi></msub><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi><mi mathvariant="normal">T</mi></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi><mi mathvariant="normal">T</mi></msubsup><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow><mo>.</mo></math> </ephtml> We assume that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mtext mathvariant="italic">ip</mtext></mrow></msub></mrow></math> </ephtml> are conditionally independent given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> and the conditional density function of <emph>Y<subs>ij</subs></emph> given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> belongs to a canonical exponential family</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mrow><mo stretchy="true">{</mo><mrow><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">ψ</mi></mrow><mrow><mi>j</mi><mn>1</mn></mrow><mi mathvariant="normal">T</mi></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mrow><mi>b</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">ψ</mi></mrow><mrow><mi>j</mi><mn>1</mn></mrow><mi mathvariant="normal">T</mi></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>a</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><msub><mrow><mi>c</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">}</mo></mrow><mo>,</mo></mrow></math> </ephtml> </p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>a</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo>,</mo><mtext /><msub><mrow><mi>b</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>c</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> are some known functions,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">ψ</mi></mrow><mrow><mi>j</mi><mn>1</mn></mrow></msub></mrow></math> </ephtml> is a vector of unknown regression parameters, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi><mn>2</mn></mrow></msub></mrow></math> </ephtml> is either known or unknown, depending on the distribution considered. For example, if <emph>Y<subs>ij</subs></emph> follows the normal distribution with mean</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>μ</mi><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">ψ</mi></mrow><mrow><mi>j</mi><mn>1</mn></mrow><mi mathvariant="normal">T</mi></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mo>=</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi><mn>2</mn></mrow></msub></mrow><mo>,</mo></math> </ephtml> then</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>a</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>,</mo><mtext /><msub><mrow><mi>b</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>c</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mo>{</mo><msup><mrow><mi>y</mi></mrow><mn>2</mn></msup><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo /><mtext mathvariant="normal">log</mtext><mo /><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>}</mo><mo>/</mo><mn>2</mn></mrow><mo>.</mo></math> </ephtml> If <emph>Y<subs>ij</subs></emph> is a binary variable following the Bernoulli distribution with a success probability of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">ψ</mi></mrow><mrow><mi>j</mi><mn>1</mn></mrow><mi mathvariant="normal">T</mi></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> then</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>a</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>b</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo /><mtext mathvariant="normal">log</mtext><mo /><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>e</mi></mrow><mi>p</mi></msup><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> In the proposed SEM framework, model (<reflink idref="bib1" id="ref38">1</reflink>) is referred to as the structural model that connects the latent variables. Model (<reflink idref="bib2" id="ref39">2</reflink>) serves as the measurement model characterizing the relationships between the latent and observed variables.</p> <p>Let <emph>T<subs>i</subs></emph> denote the failure time of interest, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> be another</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi><mo>×</mo><mn>1</mn></mrow></math> </ephtml> vector of observed covariates associated with <emph>T<subs>i</subs></emph> for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow><mo>.</mo></math> </ephtml> For each <emph>i</emph>, we assume that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> are independent of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> conditional on</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> as Wong et al. ([<reflink idref="bib34" id="ref40">34</reflink>]). The semiparametric factor-augmented transformation models specify that the cumulative hazard function of <emph>T<subs>i</subs></emph> conditional on</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> takes the form</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi><mrow><mo stretchy="true">{</mo><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">}</mo></mrow><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib3" id="ref41">3</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> is a prespecified increasing function with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mtext /><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is an unspecified cumulative baseline hazard function, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">β</mi><mo>,</mo><mtext /><mi mathvariant="bold-italic">ϑ</mi><mo>,</mo><mtext /><mi mathvariant="bold-italic">α</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">γ</mi></mrow></math> </ephtml> are the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi><mo>×</mo><mn>1</mn><mo>,</mo><mtext /><mi>p</mi><mo>×</mo><mn>1</mn><mo>,</mo><mtext /><mi>s</mi><mo>×</mo><mn>1</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi><mo>×</mo><mn>1</mn></mrow></math> </ephtml> vectors of unknown regression parameters, respectively. The class of models (<reflink idref="bib3" id="ref42">3</reflink>) is flexible and includes many commonly used semiparametric models as its special cases. For instance, we obtain the factor-augmented PH and PO models by specifying <emph>G</emph>(<emph>x</emph>) = <emph>x</emph> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo /><mtext mathvariant="normal">log</mtext><mo /><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> respectively.</p> <p>We consider the general or mixed-case interval-censored data where the <emph>i</emph>th individual undergoes <emph>J<subs>i</subs></emph> observation or monitoring times for examining the failure event occurrence. Denoted by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mn>1</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><msub><mrow><mi>J</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></math> </ephtml> the <emph>J<subs>i</subs></emph> monitoring times of the <emph>i</emph>th individual, and let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><msub><mrow><mi>J</mi></mrow><mi>i</mi></msub><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>∞</mi></mrow><mo>.</mo></math> </ephtml> For the <emph>i</emph>th individual, let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><msub><mrow><mi>J</mi></mrow><mi>i</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> We assume that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is conditionally independent of <emph>T<subs>i</subs></emph> given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow><mo>.</mo></math> </ephtml> For</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>J</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> define</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mo>△</mo></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>≤</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>I</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> is the indicator function. Define</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo>=</mo><mtext mathvariant="normal">max</mtext><mo>{</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>:</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>,</mo><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>J</mi></mrow><mi>i</mi></msub><mo>}</mo></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>=</mo><mtext mathvariant="normal">min</mtext><mo>{</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>:</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>≤</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>J</mi></mrow><mi>i</mi></msub><mo>+</mo><mn>1</mn><mo>}</mo></mrow><mo>.</mo></math> </ephtml> That is,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">]</mo></mrow></math> </ephtml> is the smallest interval that contains <emph>T<subs>i</subs></emph>. Clearly, <emph>L<subs>i</subs></emph> = 0 indicates that <emph>T<subs>i</subs></emph> is left censored, while</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>=</mo><mi>∞</mi></mrow></math> </ephtml> implies that <emph>T<subs>i</subs></emph> is right censored.</p> <p>Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mo>{</mo><mo>·</mo><mo>}</mo></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> denote the density or conditional density function of the subscript variable or variables. For example,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>F</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">x</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>F</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">x</mi></mrow><mo>.</mo></math> </ephtml> To derive the observed data likelihood, we first notice that the joint density function</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> can be decomposed as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>f</mi></mrow><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi mathvariant="bold-italic">U</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>T</mi><mo>,</mo><mi mathvariant="bold-italic">U</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><msub><mrow><mi>f</mi></mrow><mi>T</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">U</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mrow><mo>∫</mo><mrow><msub><mrow><mi>f</mi></mrow><mi>T</mi></msub></mrow></mrow><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo /><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">U</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mrow><mo>∫</mo><mrow><msub><mrow><mi>f</mi></mrow><mi>T</mi></msub></mrow></mrow><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Z</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo /><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">U</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mrow><mo>∫</mo><mrow><msub><mrow><mi>f</mi></mrow><mi>T</mi></msub></mrow></mrow><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">X</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Z</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mtd></mtr><mtr><mtd><mo>×</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">U</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mrow><mo>∫</mo><mrow><msub><mrow><mi>f</mi></mrow><mi>T</mi></msub></mrow></mrow><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mtd></mtr><mtr><mtd><mo>×</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">U</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>where the second equality follows from the assumption that <emph>T<subs>i</subs></emph> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> are conditionally independent given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> and the fifth equality follows from the assumption that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is independent of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> conditional on</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow><mo>.</mo></math> </ephtml> Based on the above derivation, the observed data likelihood function is proportional to</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mrow><mrow><mo>∫</mo></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo>{</mo><mi>Λ</mi><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>}</mo><mo>}</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo>{</mo><mi>Λ</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>}</mo><mo>}</mo><mo stretchy="true">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><mo>×</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> (<reflink idref="bib4" id="ref43">4</reflink>)</p> <hd id="AN0182192620-5">2.2. Model Identifiability</hd> <p>In what follows, notations without subscript <emph>i</emph> denote their population analogs. To ensure the model identifiability, we adopt the two rules elaborated in Wong et al. ([<reflink idref="bib34" id="ref44">34</reflink>]): (i) the latent variables attached to <emph>T</emph> can be treated as observed if <emph>T</emph> depends on at least one observed covariate; (ii) if each latent variable has a separate continuous indicator and the distributions of the latent variables and their indicators are identifiable, then the whole model is identifiable.</p> <p>To illustrate the above rules, we consider two examples of SEM depicted in Figure 1. In the first SEM depicted by the left plot of Figure 1, <emph>T</emph> denotes a failure time following the PH model with the covariate <emph>X</emph> and latent variable <emph>η</emph>. If the regression parameter for <emph>η</emph> is fixed to one,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></math> </ephtml> and the regression parameter for <emph>X</emph> is nonzero, then one has a PH frailty model with observed and latent variables. By Theorem 3 of Kortram et al. ([<reflink idref="bib13" id="ref45">13</reflink>]), the model for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is identifiable and <emph>η</emph> can be treated as observed when identifying the PH model. In the second SEM depicted by the right plot of Figure 1, Y<subs>1</subs>, <emph>Y</emph><subs>2</subs> and <emph>Y</emph><subs>3</subs> are three conditionally independent and normally distributed manifest variables of <emph>η</emph>, <emph>T</emph> is the failure time formulated by the PH model with the latent variable <emph>η</emph>. To ensure that the model for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mn>3</mn></msub><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is identifiable, it is common to assume that the regression parameter relating <emph>Y</emph><subs>1</subs> to <emph>η</emph> is one (Bollen, [<reflink idref="bib2" id="ref46">2</reflink>]; Wong et al., [<reflink idref="bib34" id="ref47">34</reflink>]). With <emph>Y</emph><subs>1</subs> serving as an indicator of <emph>η</emph>, the second rule implies that the remaining parameters are identifiable.</p> <p>Graph: Figure 1. Two examples of SEM for illustrating the identifiability rules. The left plot indicates an SEM containing one latent variable η, one observed covariate X and one failure time T. The right plot indicates an SEM including one latent variable η, one failure time T and three conditionally independent observed variables (Y1,Y2,Y3).</p> <p>Graph: Figure 2. The assumed SEM in the simulation. The SEM consists of three latent variables (η1, η2, η3), nine observed manifest covariates (Y1,...,Y9), five observed covariates (X1,...,X5), and a failure time T regressed on η3 and (X1,...,X5).</p> <p>DIAGRAM: Figure 3. Path diagram of the used SEM with analysis results for the ADNI data under the optimal model (r = 0.9). The results include the parameter estimates and their standard error estimates given in brackets.</p> <p>In our work, as Wong et al. ([<reflink idref="bib34" id="ref48">34</reflink>]) and others, we set the regression parameter for each latent variable <emph>η</emph> and one of its manifest variables to one. This condition ensures the identifiability of the non-survival submodel. Consequently, the whole SEM is identifiable by the second rule.</p> <hd id="AN0182192620-6">3. Computational Method</hd> <p>To estimate models (<reflink idref="bib1" id="ref49">1</reflink>) to (<reflink idref="bib3" id="ref50">3</reflink>), we resort to the NPMLE approach, which regards</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> as a step function taking jumps at unique observation times. Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>c</mi></mrow><mn>1</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>c</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub></mrow></msub></mrow></math> </ephtml> denote ordered distinct observation times and <emph>λ<subs>k</subs></emph> be the nonnegative jump size of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> at <emph>c<subs>k</subs></emph> with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub></mrow><mo>,</mo></math> </ephtml> where <emph>K<subs>n</subs></emph> is an integer determined by the observed data. Define</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">λ</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>λ</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow><mo>.</mo></math> </ephtml> Then one can approximate</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>Λ</mi></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><mi>t</mi></mrow></munder></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub></mrow></mrow></math> </ephtml> and the likelihood (2.1) becomes</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow /><mrow><mo>∫</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mrow><mo stretchy="true">(</mo><mrow><mo>−</mo><mi>G</mi><mrow><mo stretchy="true">{</mo><mrow><msub><mrow><mi>Λ</mi></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">}</mo></mrow></mrow><mo stretchy="true">)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><mo>−</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mrow><mo stretchy="true">(</mo><mrow><mo>−</mo><mi>G</mi><mrow><mo stretchy="true">{</mo><mrow><msub><mrow><mi>Λ</mi></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">}</mo></mrow></mrow><mo stretchy="true">)</mo></mrow><mo stretchy="true">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><mo>×</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> (<reflink idref="bib5" id="ref51">5</reflink>)</p> <p>Notably, performing direct maximization of the likelihood (<reflink idref="bib5" id="ref52">5</reflink>) is challenging since it has an intractable form and involves multiple integrations. We develop an EM algorithm that introduces three sets of latent variables to facilitate the maximization. First, note that the transformation function <emph>G</emph>(<emph>x</emph>) can be induced from the Laplace transformation as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><mo>=</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>x</mi><mi>μ</mi></mrow></msup></mrow></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>;</mo><mi>r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>μ</mi></mrow><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>;</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is the density function of the frailty <emph>μ</emph> with parameter <emph>r</emph> (Kosorok et al., [<reflink idref="bib14" id="ref53">14</reflink>]). If</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>;</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is the density function of a gamma random variable with mean 1 and variance <emph>r</emph>, we have</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo /><mtext mathvariant="normal">log</mtext><mo /><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mtext mathvariant="italic">rx</mtext><mo stretchy="false">)</mo><mo>/</mo><mi>r</mi><mo /><mo stretchy="false">(</mo><mi>r</mi><mo>≥</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> the logarithmic transformation family in which <emph>r</emph> = 0 means <emph>G</emph>(<emph>x</emph>) = <emph>x</emph>. Through this transformation, the class of models (<reflink idref="bib3" id="ref54">3</reflink>) can be expressed as the PH frailty models, and the likelihood (<reflink idref="bib5" id="ref55">5</reflink>) is equivalent to</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow /><mo>∫∫</mo></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><msub><mrow><mi>Λ</mi></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>}</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><mo>−</mo><mo /><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><msub><mrow><mi>Λ</mi></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><mo /><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>}</mo><mo stretchy="true">]</mo><mo /><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>;</mo><mi>r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub></mrow><mrow><msub><mrow><mo /><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> (<reflink idref="bib6" id="ref56">6</reflink>)</p> <p>By treating</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> s and <emph>μ<subs>i</subs></emph>s as observable, the augmented-data likelihood is given by</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow /><mtext /><mrow><mo stretchy="true">[</mo><mrow><mtext mathvariant="normal">exp</mtext><mo /><mrow><mo stretchy="true">{</mo><mrow><mo>−</mo><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow></mrow><mo stretchy="true">}</mo></mrow><mo>−</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mrow><mo stretchy="true">{</mo><mrow><mo>−</mo><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow></mrow><mo stretchy="true">}</mo></mrow></mrow><mo>]</mo></mrow><mo>×</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>;</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo /><mo>|</mo><mo /><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo /><mo>|</mo><mo /><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib7" id="ref57">7</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub></mrow><mo>.</mo></math> </ephtml> Next, we introduce a set of new independent latent variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo>;</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub><mo>}</mo></mrow><mo>,</mo></math> </ephtml> where <emph>W<subs>ik</subs></emph> is a Poisson random variable with mean <emph>ω<subs>ik</subs></emph>. Then, the likelihood (<reflink idref="bib7" id="ref58">7</reflink>) is equivalent to</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mrow><mo stretchy="true">(</mo><mrow><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow><mo>=</mo><mn>0</mn></mrow><mo stretchy="true">)</mo></mrow><mi>P</mi><msup><mrow><mrow><mrow><mo stretchy="true">(</mo><mrow><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo><</mo><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow><mo>></mo><mn>0</mn></mrow><mo stretchy="true">)</mo></mrow></mrow></mrow><mrow><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo></mrow></msup><mo>×</mo><mo /><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>;</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mrow><mo /><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo /><mo>|</mo><mo /><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><msub><mrow><mo /><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo /><mo>|</mo><mo /><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> (<reflink idref="bib8" id="ref59">8</reflink>)</p> <p>Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo>|</mo><msub><mrow><mi>ω</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> denote the probability mass function of <emph>W<subs>ik</subs></emph> with mean <emph>ω<subs>ik</subs></emph>. Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mi mathvariant="normal">T</mi></msup><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">θ</mi></mrow></math> </ephtml> is a vector containing all finite-dimensional parameters in models (<reflink idref="bib1" id="ref60">1</reflink>) to (<reflink idref="bib3" id="ref61">3</reflink>). By further treating <emph>W<subs>ik</subs></emph>s as observable, we would have the following complete data likelihood</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mrow><mi>L</mi></mrow><mi>c</mi></msub><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow /><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub></mrow></munderover><mrow /><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo>|</mo><msub><mrow><mi>ω</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>;</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow /><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub></mrow></munderover><mrow /><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><mo>−</mo><msub><mrow><mi>ω</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo><mfrac><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow></msubsup></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo>!</mo></mrow></mfrac><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>;</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib9" id="ref62">9</reflink>)</p> <p>where we know from likelihood (<reflink idref="bib8" id="ref63">8</reflink>) that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><munder><mo>∑</mo><mrow><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub></mrow><mo>≤</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo><</mo><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow><mo>></mo><mn>0</mn></mrow></math> </ephtml> if</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow><mo>=</mo><mn>0</mn></mrow></math> </ephtml> if</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>=</mo><mi>∞</mi></mrow><mo>.</mo></math> </ephtml> In other words, under this Poisson variable framework, the observed data for the <emph>i</emph>th individual, denoted by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="script">O</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> consist of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mi>i</mi></msub><mo>=</mo><mn>0</mn><mo>}</mo></mrow></math> </ephtml> if</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>=</mo><mi>∞</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mi>i</mi></msub><mo>=</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>B</mi></mrow><mi>i</mi></msub><mo>></mo><mn>0</mn><mo>}</mo></mrow></math> </ephtml> if</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi></mrow><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>A</mi></mrow><mi>i</mi></msub><mo>=</mo><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>B</mi></mrow><mi>i</mi></msub><mo>=</mo><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo><</mo><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub></mrow></mrow><mo>.</mo></math> </ephtml> </p> <p>In the E-step of the algorithm, we take the conditional expectation of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo /><mtext mathvariant="normal">log</mtext><mo /><msub><mrow><mi>L</mi></mrow><mi>c</mi></msub><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow></math> </ephtml> with respect to all latent variables and have</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Q</mi><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo>;</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>Q</mi></mrow><mn>1</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">β</mi><mo>,</mo><mi mathvariant="bold-italic">ϑ</mi><mo>,</mo><mi mathvariant="bold-italic">α</mi><mo>,</mo><mi mathvariant="bold-italic">γ</mi><mo>,</mo><mi mathvariant="bold-italic">λ</mi><mo>;</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ψ</mi><mo>,</mo><mi mathvariant="bold-italic">ν</mi><mo>;</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> (<reflink idref="bib10" id="ref64">10</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mrow><mi>Q</mi></mrow><mn>1</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">β</mi><mo>,</mo><mi mathvariant="bold-italic">ϑ</mi><mo>,</mo><mi mathvariant="bold-italic">α</mi><mo>,</mo><mi mathvariant="bold-italic">γ</mi><mo>,</mo><mi mathvariant="bold-italic">λ</mi><mo>;</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub></mrow></munderover><mo stretchy="true">[</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mo /><mtext mathvariant="normal">log</mtext><mo /><mo stretchy="false">(</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><mo>−</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mrow><mo stretchy="true">{</mo><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mrow><mo stretchy="true">(</mo><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="true">)</mo></mrow><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="true">}</mo></mrow><mo stretchy="true">]</mo><mo>,</mo></mrow></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mrow><mi>Q</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ψ</mi><mo>,</mo><mi mathvariant="bold-italic">ν</mi><mo>;</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi mathvariant="normal">E</mi><mrow><mo stretchy="true">{</mo><mrow><mo /><mtext mathvariant="normal">log</mtext><mo stretchy="true">[</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><mo stretchy="true">]</mo></mrow><mo stretchy="true">}</mo></mrow><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi mathvariant="normal">E</mi><mrow><mo stretchy="true">{</mo><mrow><mo /><mtext mathvariant="normal">log</mtext><mo stretchy="true">[</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mo stretchy="true">]</mo></mrow><mo stretchy="true">}</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math> </ephtml> </p> <p>and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>Q</mi></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math> </ephtml> is a function of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup></mrow></math> </ephtml> but is free of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow><mo>.</mo></math> </ephtml> In the above,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup></mrow></math> </ephtml> denotes the update of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></math> </ephtml> at the <emph>m</emph>th iteration with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow><mo>.</mo></math> </ephtml> For ease of presentation, we omit the conditional arguments, including the observed data and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mo>,</mo></math> </ephtml> in all the conditional expectations.</p> <p>Define</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub></mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>and</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mrow><munder><mo>∑</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub></mrow></munder></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub></mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>.</mo></mrow></math> </ephtml> </p> <p>Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">d</mi><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mo>.</mo></math> </ephtml> By applying the law of iterated expectations and Bayesian theorem, one can obtain the following conditional expectations</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></msub><mo>{</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo>|</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>}</mo></mtd></mtr><mtr><mtd /></mtr><mtr><mtd><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold">η</mi></mrow><mi>i</mi></msub></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo><msup><mrow><mi>G</mi></mrow><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo><mo>−</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo></mrow></mfrac></mrow><mo stretchy="true">]</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">]</mo></mrow><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mi>k</mi></msub><mo>></mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">]</mo></mrow><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>=</mo><mi>∞</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mi>k</mi></msub><mo>></mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold">η</mi></mrow><mi>i</mi></msub></mrow></msub><mrow><mo stretchy="true">{</mo><mrow><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">}</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold">η</mi></mrow><mi>i</mi></msub></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo><msup><mrow><mi>G</mi></mrow><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo><mo>−</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo></mrow></mfrac></mrow><mo stretchy="true">]</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo><</mo><msub><mrow><mi>t</mi></mrow><mi>k</mi></msub><mo>≤</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">]</mo></mrow><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mi>k</mi></msub><mo>></mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">]</mo></mrow><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>=</mo><mi>∞</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mi>k</mi></msub><mo>></mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo><mi>G</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>−</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo><mi>G</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo><mo>−</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>}</mo></mrow></mfrac><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mo>+</mo><mi>G</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo>=</mo><mi>∞</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr></mtable></math> </ephtml> </p> <p>and</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">E</mi><mo>{</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>}</mo><mo>=</mo><mfrac><mrow><mrow><mo>∫</mo></mrow><mi>h</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow><mrow><mrow><mo>∫</mo></mrow><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></mfrac><mo>,</mo></mrow></math> </ephtml> </p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>h</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> denotes an arbitrary function of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>{</mo><mo /><mtext mathvariant="normal">exp</mtext><mo stretchy="true">[</mo><mo>−</mo><mi>G</mi><mo>{</mo><msub><mrow><mi>Λ</mi></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>}</mo><mo stretchy="true">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo><</mo><mi>∞</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo stretchy="true">[</mo><mo>−</mo><mi>G</mi><mo>{</mo><msub><mrow><mi>Λ</mi></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>}</mo><mo stretchy="true">]</mo><mo>}</mo></mtd></mtr><mtr><mtd><mo>×</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo stretchy="false">)</mo><msub><mrow><mi>f</mi></mrow><mi mathvariant="bold-italic">η</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo>|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>;</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>We use the multivariate Gaussian–Hermite quadrature to calculate the integral of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> that has no explicit form. Detailed derivations of the above conditional expectations can be found in Appendix A of the Supplementary Material.</p> <p>In the M-step, solving</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>∂</mo><msub><mrow><mi>Q</mi></mrow><mn>1</mn></msub><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo>,</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mo>∂</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> leads to a closed-form solution for <emph>λ<subs>k</subs></emph>, which takes the form</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>λ</mi></mrow><mi>k</mi><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo>{</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo>}</mo></mrow></mfrac><mo>,</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub><mo>.</mo></mrow></math> </ephtml> </p> <p>By plugging</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>λ</mi></mrow><mi>k</mi><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> </ephtml> s into</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>Q</mi></mrow><mn>1</mn></msub><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo>;</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> the score equations for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">β</mi><mo>,</mo><mtext /><mi mathvariant="bold-italic">ϑ</mi><mo>,</mo><mtext /><mi mathvariant="bold-italic">α</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">γ</mi></mrow></math> </ephtml> are</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo><mrow><mo stretchy="true">{</mo><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo stretchy="true">]</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo stretchy="true">]</mo></mrow></mfrac></mrow><mo stretchy="true">}</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo><mrow><mo stretchy="true">{</mo><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo stretchy="true">]</mo><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo stretchy="true">]</mo></mrow></mfrac></mrow><mo stretchy="true">}</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo><mrow><mo stretchy="true">{</mo><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo stretchy="true">]</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo stretchy="true">]</mo></mrow></mfrac></mrow><mo stretchy="true">}</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>and</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo /><mrow><mo stretchy="true">{</mo><mrow><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>‐</mo><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mtext mathvariant="italic">ik</mtext></mrow></msub><mo stretchy="false">)</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo stretchy="true">]</mo></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi>i</mi></msub><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">E</mi><mo stretchy="true">[</mo><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mrow><mi>μ</mi></mrow><mi>i</mi></msub><mo stretchy="true">]</mo></mrow></mfrac></mrow><mo stretchy="true">}</mo></mrow><mo>=</mo><mn>0.</mn></mrow></math> </ephtml> </p> <p>In addition, we update the remaining parameters by solving</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>∂</mo><msub><mrow><mi>Q</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ψ</mi><mo>,</mo><mi mathvariant="bold-italic">ν</mi><mo>;</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mo>∂</mo><mi mathvariant="bold-italic">ψ</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>∂</mo><msub><mrow><mi>Q</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ψ</mi><mo>,</mo><mi mathvariant="bold-italic">ν</mi><mo>;</mo><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mo>∂</mo><mi mathvariant="bold-italic">ν</mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math> </ephtml> If no closed-form solutions are available, one can obtain the parameter updates by applying the one-step Newton-Raphson algorithm to the above estimating equations. The proposed EM algorithm is insensitive to the choices of the initial value</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><mo>.</mo></math> </ephtml> In implementing the proposed algorithm, one can set the initial value of each finite-dimensional parameter to 0 and the initial value of each <emph>λ<subs>k</subs></emph> to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>K</mi></mrow><mi>n</mi></msub></mrow><mo>.</mo></math> </ephtml> The initial values of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ψ</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ν</mi></mrow></math> </ephtml> are set to the identity matrices with appropriate dimensions. Moreover, one can monitor the algorithm's convergence through a commonly used stopping criterion</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><munder><mo>∑</mo><mrow><mi>j</mi><mo>:</mo><msub><mrow><mi>ζ</mi></mrow><mi>j</mi></msub><mo>∈</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></munder></mrow><mo stretchy="true">|</mo><msubsup><mrow><mi>ζ</mi></mrow><mi>j</mi><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mrow><mi>ζ</mi></mrow><mi>j</mi><mrow><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>|</mo><mo><</mo><mi>δ</mi><mo>,</mo></mrow></math> </ephtml> where <emph>δ</emph> is a predetermined small threshold (e.g., 0.001). The proposed method is implemented under the Rcpp environment in R, which can accelerate the computation.</p> <hd id="AN0182192620-7">4. Asymptotic Properties</hd> <p>Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mrow><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow></math> </ephtml> denote the proposed nonparametric maximum likelihood estimator of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ζ</mi><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>Λ</mi><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> For a column vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo></math> </ephtml> let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>|</mo><mo>|</mo><mi mathvariant="bold-italic">a</mi><mo>|</mo><msub><mrow><mo>|</mo></mrow><mn>2</mn></msub></mrow></math> </ephtml> be the <emph>L</emph><subs>2</subs>-norm of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo></math> </ephtml> which is defined as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">a</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>.</mo></math> </ephtml> To establish the asymptotic properties of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo>,</mo></math> </ephtml> we use the following regularity conditions.</p> <p></p> <ulist> <item> (C1) The true value of</item> </ulist> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>,</mo></math> </ephtml> denoted by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mn>0</mn></msub><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mn>0</mn><mi mathvariant="normal">T</mi></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mn>0</mn><mi mathvariant="normal">T</mi></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mn>0</mn><mi mathvariant="normal">T</mi></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mn>0</mn><mi mathvariant="normal">T</mi></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">ψ</mi></mrow><mn>0</mn><mi mathvariant="normal">T</mi></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">ν</mi></mrow><mn>0</mn><mi mathvariant="normal">T</mi></msubsup><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow><mo>,</mo></math> </ephtml> belongs to a compact product set</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Θ</mi><mo>≡</mo><msub><mrow><mi mathvariant="script">B</mi></mrow><mn>1</mn></msub><mo>×</mo><msub><mrow><mi mathvariant="script">B</mi></mrow><mn>2</mn></msub><mo>×</mo><msub><mrow><mi mathvariant="script">B</mi></mrow><mn>3</mn></msub><mo>×</mo><mi mathvariant="script">G</mi><mo>×</mo><msub><mrow><mi mathvariant="script">C</mi></mrow><mn>1</mn></msub><mo>×</mo><msub><mrow><mi mathvariant="script">C</mi></mrow><mn>2</mn></msub></mrow><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="script">B</mi></mrow><mn>1</mn></msub><mo>⊂</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>r</mi></msup><mo>,</mo><mtext /><msub><mrow><mi mathvariant="script">B</mi></mrow><mn>2</mn></msub><mo>⊂</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>p</mi></msup><mo>,</mo><mtext /><msub><mrow><mi mathvariant="script">B</mi></mrow><mn>3</mn></msub><mo>⊂</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>s</mi></msup><mo>,</mo><mtext /><mi mathvariant="script">G</mi><mo>⊂</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>q</mi></msup></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="script">C</mi></mrow><mn>1</mn></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="script">C</mi></mrow><mn>2</mn></msub></mrow></math> </ephtml> are the compact sets in the domains of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ψ</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ν</mi></mrow><mo>,</mo></math> </ephtml> respectively. The true value of Λ, denoted by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>Λ</mi></mrow><mn>0</mn></msub></mrow><mo>,</mo></math> </ephtml> is continuously differentiable with the positive first derivative in</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">[</mo><mi>c</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></mrow><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">[</mo><mi>c</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></mrow></math> </ephtml> is the union of the supports of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>U</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mi>J</mi></msub><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> <emph>c</emph> is a small positive constant and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>τ</mi><mo><</mo><mi>∞</mi></mrow><mo>.</mo></math> </ephtml> </p> <p></p> <ulist> <item> (C2) The covariate vectors</item> </ulist> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">X</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">Z</mi></mrow></math> </ephtml> are bounded with probability one.</p> <p></p> <ulist> <item> (C3) The number of monitoring times <emph>J</emph> is positive with</item> </ulist> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo><</mo><mi>∞</mi></mrow><mo>.</mo></math> </ephtml> The conditional probability</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>U</mi></mrow><mi>J</mi></msub><mo>=</mo><mi>τ</mi><mo>|</mo><mi>J</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is greater than some positive constant <emph>b</emph>. For</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>J</mi><mo>−</mo><mn>1</mn></mrow><mo>,</mo></math> </ephtml> let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>g</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> denote the conditional joint density of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>U</mi></mrow><mi>j</mi></msub><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>J</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> For some positive constant <emph>c</emph>,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>g</mi></mrow><mi>j</mi></msub><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> has continuous second-order partial derivatives with respect to <emph>u</emph> and <emph>v</emph> when</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>v</mi><mo>−</mo><mi>u</mi><mo>≥</mo><mi>c</mi></mrow><mo>,</mo></math> </ephtml> and is continuously differentiable with respect to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">X</mi><mo>,</mo><mtext /><mi mathvariant="bold-italic">Y</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mo>.</mo></math> </ephtml> In addition,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>U</mi></mrow><mi>j</mi></msub><mo>≥</mo><mi>c</mi><mo>|</mo><mi>J</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow></math> </ephtml> for any</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>J</mi></mrow><mo>.</mo></math> </ephtml> </p> <p></p> <ulist> <item> (C4) The transformation function <emph>G</emph> is twice continuously differentiable on</item> </ulist> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">d</mi><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>></mo><mn>0</mn></mrow><mo>.</mo></math> </ephtml> Also,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi mathvariant="italic">ρ</mi></mrow><mn>0</mn></msub></mrow></msup><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><mo><</mo><mi>∞</mi></mrow></math> </ephtml> as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>→</mo><mi>∞</mi></mrow></math> </ephtml> for some positive constant <emph>ρ</emph><subs>0</subs>. In addition,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></math> </ephtml> is uniformly bounded in</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo>{</mo><mo>−</mo><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mn>1</mn></msub><msup><mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mo>−</mo><msub><mrow><mi mathvariant="italic">ρ</mi></mrow><mn>1</mn></msub></mrow></msup></mrow></math> </ephtml> for some <emph>a</emph><subs>1</subs> and <emph>ρ</emph><subs>1</subs>.</p> <p></p> <ulist> <item> (C5) Consider two sets of parameters</item> </ulist> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ψ</mi><mo>,</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ψ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo>,</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ν</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi><mo>|</mo><mi mathvariant="bold-italic">Z</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the joint density of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">Y</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">η</mi></mrow></math> </ephtml> given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mo>.</mo></math> </ephtml> For any</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ψ</mi><mo>,</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ψ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo>,</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ν</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow></math> </ephtml> in the parameter space,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi><mo>|</mo><mi mathvariant="bold-italic">Z</mi><mo>;</mo><mi mathvariant="bold-italic">ψ</mi><mo>,</mo><mi mathvariant="bold-italic">ν</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi><mo>|</mo><mi mathvariant="bold-italic">Z</mi><mo>;</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ψ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo>,</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ν</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow></math> </ephtml> for all</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">Z</mi><mo>,</mo><mtext /><mi mathvariant="bold-italic">Y</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">η</mi></mrow></math> </ephtml> implies that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ψ</mi><mo>=</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ψ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">ν</mi><mo>=</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ν</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow><mo>.</mo></math> </ephtml> </p> <p></p> <ulist> <item> (C6) For any</item> </ulist> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>c</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></mrow><mo>,</mo></math> </ephtml> if there exist constant vectors</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">b</mi><mo>,</mo><mtext /><mi mathvariant="bold-italic">c</mi><mo>,</mo><mi mathvariant="bold-italic">d</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">e</mi></mrow><mo>,</mo></math> </ephtml> and a deterministic function</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> such that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">b</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">X</mi><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">Y</mi><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">d</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">Z</mi><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">η</mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></math> </ephtml> then</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>|</mo><mo>|</mo><mi mathvariant="bold-italic">b</mi><mo>|</mo><msub><mrow><mo>|</mo></mrow><mn>2</mn></msub><mo>=</mo><mo>|</mo><mo>|</mo><mi mathvariant="bold-italic">c</mi><mo>|</mo><msub><mrow><mo>|</mo></mrow><mn>2</mn></msub><mo>=</mo><mo>|</mo><mo>|</mo><mi mathvariant="bold-italic">d</mi><mo>|</mo><msub><mrow><mo>|</mo></mrow><mn>2</mn></msub><mo>=</mo><mo>|</mo><mo>|</mo><mi mathvariant="bold-italic">e</mi><mo>|</mo><msub><mrow><mo>|</mo></mrow><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></math> </ephtml> for any</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>c</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></mrow><mo>.</mo></math> </ephtml> </p> <p></p> <ulist> <item> (C7) Let</item> </ulist> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">κ</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">ψ</mi></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">ν</mi></mrow><mi mathvariant="normal">T</mi></msup><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow><mo>,</mo></math> </ephtml> if there exist a function</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>t</mi><mo>;</mo><mi mathvariant="bold-italic">η</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and a constant vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">b</mi></mrow></math> </ephtml> such that</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>∫</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mrow><mo stretchy="true">[</mo><mrow><mo>−</mo><mi>G</mi><mo>{</mo><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">X</mi><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ϑ</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">Y</mi><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">Z</mi><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">η</mi><mo stretchy="false">)</mo><mo>}</mo></mrow><mo stretchy="true">]</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><mo>×</mo><mrow><mo stretchy="true">{</mo><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>t</mi><mo>;</mo><mi mathvariant="bold-italic">η</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">b</mi></mrow><mi mathvariant="normal">T</mi></msup><mfrac><mrow><mo>∂</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi><mo>|</mo><mi mathvariant="bold-italic">Z</mi><mo>;</mo><mi mathvariant="bold-italic">κ</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">κ</mi></mrow></mfrac></mrow><mo stretchy="true">}</mo></mrow><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">η</mi><mo>|</mo><mi mathvariant="bold-italic">Z</mi><mo>;</mo><mi mathvariant="bold-italic">κ</mi><mo stretchy="false">)</mo><mi>d</mi><mi mathvariant="bold-italic">η</mi><mo>=</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>with probability one for any</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>c</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></mrow><mo>,</mo></math> </ephtml> then</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>t</mi><mo>;</mo><mi mathvariant="bold-italic">η</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">b</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>c</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></mrow><mo>.</mo></math> </ephtml> </p> <p>Conditions (C1)–(C3) are standard and commonly adopted in the literature of interval-censored data analysis (Zeng et al., [<reflink idref="bib38" id="ref65">38</reflink>], [<reflink idref="bib36" id="ref66">36</reflink>]). Condition (C4) holds for the logarithmic transformation function family (Zeng et al., [<reflink idref="bib37" id="ref67">37</reflink>]). Condition (C5) pertains to the parameter identifiability of Section (2.2), which is derived in more detail by Wong et al. ([<reflink idref="bib34" id="ref68">34</reflink>]). Condition (C6) holds if the matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi mathvariant="normal">T</mi></msup><mo stretchy="false">]</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi mathvariant="normal">T</mi></msup><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> is nonsingular. Condition (C7) is used to show that the Fisher information matrix along any submodel at the true parameter values is positive definite or nonsingular (Zeng et al., [<reflink idref="bib36" id="ref69">36</reflink>]). The asymptotic properties of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ζ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> are described in the following two theorems with the detailed proofs presented in Appendix B of the Supplementary Material. Some useful lemmas are given in Appendix C of the Supplementary Material.</p> <hd id="AN0182192620-8">Theorem 1.</hd> <p>Under the regularity conditions (C1)–(C5), we have</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>|</mo><mo>|</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mn>0</mn></msub><mo>|</mo><mo>|</mo><mo>+</mo><mrow><munder><mrow><mtext mathvariant="normal">sup</mtext></mrow><mrow><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>c</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></mrow></munder></mrow><mo>|</mo><mrow><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mrow><mi>Λ</mi></mrow><mn>0</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>|</mo><mo>→</mo><mn>0</mn></mrow></math> </ephtml> almost surely as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow><mo>.</mo></math> </ephtml> </p> <hd id="AN0182192620-9">Theorem 2.</hd> <p>Under the regularity conditions (C1)–(C7),</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mi>n</mi></msqrt><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> converges in distribution to a zero-mean multivariate normal vector with the limiting covariance matrix that attains the semiparametric efficiency bound.</p> <p>To make inference on</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mo>,</mo></math> </ephtml> one often needs to estimate the covariance matrix of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo>.</mo></math> </ephtml> However, as shown in the proof of Theorem 2, it is difficult to derive an explicit expression for the asymptotic covariance of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo>.</mo></math> </ephtml> As an alternative, following the suggestions of Zeng et al. ([<reflink idref="bib36" id="ref70">36</reflink>]) and others, we propose to adopt a numerical profile likelihood approach and estimate the covariance matrix of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><msub><mrow><mrow><mrow><mover accent="true"><mi>Σ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>,</mo></math> </ephtml> where</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mrow><mover accent="true"><mi>Σ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mi>n</mi></msub><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msup><mrow><mrow><mrow><mo stretchy="true">{</mo><mrow><msub><mrow><mrow><mrow><mrow><mfrac><mrow><mo>∂</mo><msub><mrow><mi>l</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">λ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mi mathvariant="bold-italic">θ</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow></mfrac></mrow><mo stretchy="true">|</mo></mrow></mrow></mrow><mrow><mi mathvariant="bold-italic">θ</mi><mo>=</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></msub></mrow><mo stretchy="true">}</mo></mrow></mrow></mrow><mrow><mo>⊗</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>l</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">λ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mi mathvariant="bold-italic">θ</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> is the log-likelihood function contributed by subject <emph>i</emph>,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">λ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mi mathvariant="bold-italic">θ</mi></msub><mo>=</mo><mtext mathvariant="normal">arg</mtext><msub><mrow><mrow><mtext mathvariant="normal">max</mtext></mrow></mrow><mi mathvariant="bold-italic">λ</mi></msub><mo /><mtext mathvariant="normal">log</mtext><mo /><mi>L</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi mathvariant="bold-italic">λ</mi><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>L</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi mathvariant="bold-italic">λ</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is given in (<reflink idref="bib5" id="ref71">5</reflink>). In practical implementation, given a fixed</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mtext /><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">λ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mi mathvariant="bold-italic">θ</mi></msub></mrow></math> </ephtml> can be obtained by a simplified version of the EM algorithm proposed above. Furthermore, we propose to approximate</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>∂</mo><msub><mrow><mi>l</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">λ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mi mathvariant="bold-italic">θ</mi></msub><mo stretchy="false">)</mo><mo>/</mo><mo>∂</mo><mi mathvariant="bold-italic">θ</mi><msub><mrow><mo>|</mo></mrow><mrow><mi mathvariant="bold-italic">θ</mi><mo>=</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></msub></mrow></math> </ephtml> by the first-order numerical difference.</p> <hd id="AN0182192620-10">5. Simulation Studies</hd> <p>In this section, we conducted simulations to examine the finite-sample performance of the proposed method. We first considered a setup miming the SEM used in the ADNI data analysis below. Specifically, we assumed that there exist three latent variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">η</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow><mo>,</mo></math> </ephtml> nine observed continuous manifest variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">Y</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mn>9</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> and five other observed variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">X</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>3</mn></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>4</mn></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>5</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> related to the transformation model. The structural models that relate the latent variables were specified as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub><mo>|</mo><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>ν</mi></mrow><mn>1</mn></msub><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub><mo>|</mo><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>ν</mi></mrow><mn>2</mn></msub><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>,</mo><mtext>and</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>where <emph>ν</emph><subs>1</subs> and <emph>ν</emph><subs>2</subs> are two regression parameters,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub></mrow><mn>2</mn></msubsup></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></math> </ephtml> are the variances of the error terms, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub></mrow><mn>2</mn></msubsup></mrow></math> </ephtml> is the variance of <emph>η</emph><subs>3</subs>. The true values of <emph>ν</emph><subs>1</subs> and <emph>ν</emph><subs>2</subs> were set to 0.5 and −0.5, respectively. The true values of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub></mrow><mn>2</mn></msubsup><mo>,</mo><mtext /><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub></mrow><mn>2</mn></msubsup></mrow></math> </ephtml> were all set to 0.5.</p> <p>The measurement models that relate the latent and observed variables were set to</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>Y</mi></mrow><mi>j</mi></msub><mo>|</mo><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>ψ</mi></mrow><mi>j</mi></msub><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mi>j</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mtext>for</mtext><mo /><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>Y</mi></mrow><mi>j</mi></msub><mo>|</mo><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>ψ</mi></mrow><mi>j</mi></msub><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mi>j</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mtext>for</mtext><mo /><mi>j</mi><mo>=</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mtext>and</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow /></mtd><mtd columnalign="left"><mrow><msub><mrow><mi>Y</mi></mrow><mi>j</mi></msub><mo>|</mo><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>ψ</mi></mrow><mi>j</mi></msub><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mi>j</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mtext>for</mtext><mo /><mi>j</mi><mo>=</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>9</mn><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>ψ</mi></mrow><mn>1</mn></msub><mo>...</mo><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mn>9</mn></msub></mrow></math> </ephtml> are the regression parameters, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mn>9</mn><mn>2</mn></msubsup></mrow></math> </ephtml> are the variances of the error terms. The true values of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>ψ</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mn>3</mn></msub><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mn>4</mn></msub></mrow><mo>,</mo></math> </ephtml> <emph>ψ</emph> <subs>6</subs>, <emph>ψ</emph><subs>8</subs> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>ψ</mi></mrow><mn>9</mn></msub></mrow></math> </ephtml> were set to 1. For the model identifiability consideration, we fixed <emph>ψ</emph><subs>1</subs>, <emph>ψ</emph><subs>5</subs> and <emph>ψ</emph><subs>7</subs> to 1. The true values of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mn>1</mn><mn>2</mn></msubsup></mrow></math> </ephtml> to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mn>9</mn><mn>2</mn></msubsup></mrow></math> </ephtml> were all set to 0.5. We generated <emph>X</emph><subs>1</subs> and <emph>X</emph><subs>2</subs> independently from <emph>N</emph>(0, 1), and <emph>X</emph><subs>3</subs>, <emph>X</emph><subs>4</subs> and <emph>X</emph><subs>5</subs> independently from</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="italic">Bernoulli</mtext><mo stretchy="false">(</mo><mn>0.5</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> Given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">X</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>3</mn></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>4</mn></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>5</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> and <emph>η</emph><subs>3</subs>, the cumulative hazard function of the failure time <emph>T</emph> was assumed to be</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>|</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi><mo>{</mo><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo /><mtext mathvariant="normal">exp</mtext><mo /><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mi mathvariant="normal">T</mi></msup><mi mathvariant="bold-italic">X</mi><mo>+</mo><mi>γ</mi><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo stretchy="false">)</mo><mo>}</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo /><mtext mathvariant="normal">log</mtext><mo /><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mtext mathvariant="italic">rx</mtext><mo stretchy="false">)</mo><mo>/</mo><mi>r</mi></mrow><mo>,</mo></math> </ephtml> <emph>r</emph> = 0 or 1,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo /><mtext mathvariant="normal">log</mtext><mo /><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>,</mo><mtext /><mi mathvariant="bold-italic">β</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>β</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mn>4</mn></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mn>5</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><mn>0.5</mn><mo>,</mo><mo>−</mo><mn>0.5</mn><mo>,</mo><mn>0.5</mn><mo>,</mo><mo>−</mo><mn>0.5</mn><mo>,</mo><mn>0.5</mn><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>γ</mi><mo>=</mo><mn>0.5</mn></mrow><mo>.</mo></math> </ephtml> Notably, the above model with <emph>r</emph> = 0 and 1 corresponds to the PH and PO models, respectively. The assumed SEM above was depicted in Figure 2.</p> <p>To create interval censoring, we generated a series of examination times for each subject, in which the initial examination time and the gap time between subsequently successive examination times followed the multinomial distribution on</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><mn>0.1</mn><mo>,</mo><mn>0.2</mn><mo>,</mo><mn>0.3</mn><mo>,</mo><mn>0.4</mn><mo>,</mo><mn>0.5</mn><mo>}</mo></mrow></math> </ephtml> with equal probabilities. The length of study was set to 3, beyond which no further examinations were allowed. The smallest time interval that brackets <emph>T</emph> was formed by contrasting <emph>T</emph> with the generated examination times. The sample size <emph>n</emph> was set to be 200 or 500, and 1000 replications was used. On average, we had about</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>9</mn><mi>%</mi><mo>−</mo><mn>31</mn><mi>%</mi></mrow></math> </ephtml> left-censored and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>25</mn><mi>%</mi><mo>−</mo><mn>58</mn><mi>%</mi></mrow></math> </ephtml> right-censored observations.</p> <p>Table 1 presents the simulation results with <emph>r</emph> = 0 (PH submodel), which includes the estimation bias given by the average of 1000 estimates minus the true value, the sample standard error (SSE) of the 1000 estimates, the average of the 1000 standard error estimates (SEE) and the 95% coverage probability (CP). In particular, since the parameters</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mi>j</mi><mn>2</mn></msubsup><mo /><mo stretchy="false">(</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mn>9</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mi>j</mi></msub></mrow><mn>2</mn></msubsup><mo /><mo stretchy="false">(</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> are always non-negative, we calculated their CPs using the log transformation and constructing the confidence intervals with the delta method, as in Mao and Lin ([<reflink idref="bib22" id="ref72">22</reflink>]) and references therein. The results in Table 1 indicate that, regardless of the sample size, the proposed method performs reasonably well in finite samples. All the estimation biases are virtually small, SEEs obtained by the profile likelihood method align with SSEs, and the CP values are close to the nominal level of 95%. Table 2 summarizes the simulation results with <emph>r</emph> = 1 (PO submodel) and again shows the satisfactory performance of the proposed method.</p> <p>Table 1. Simulation results for the SEM with the PH submodel (<emph>r</emph> = 0), including the estimation bias (bias), the sample standard error (SSE) of the estimates, the average of the standard error estimates (SEE), and the 95% empirical coverage probability (CP). Each row corresponds to the parameter in the model or distribution of "resp".</p> <p> <ephtml> <table><thead><tr><td /><td /><td /><td><italic>n</italic> = 200</td><td><italic>n</italic> = 500</td></tr><tr><td>Resp</td><td>Para.</td><td>True</td><td>Bias</td><td>SSE</td><td>SEE</td><td>CP</td><td>Bias</td><td>SSE</td><td>SEE</td><td>CP</td></tr></thead><tbody valign="top"><tr><td><italic>T</italic></td><td><italic>β</italic><sub>1</sub></td><td char=".">0.5</td><td char=".">0.0218</td><td char=".">0.1070</td><td char=".">0.1107</td><td char=".">97.5</td><td char=".">0.0120</td><td char=".">0.0631</td><td char=".">0.0643</td><td char=".">95.3</td></tr><tr><td /><td><italic>β</italic><sub>2</sub></td><td char=".">−0.5</td><td char=".">−0.0170</td><td char=".">0.1046</td><td char=".">0.1105</td><td char=".">96.7</td><td char=".">−0.0071</td><td char=".">0.0636</td><td char=".">0.0640</td><td char=".">95.5</td></tr><tr><td /><td><italic>β</italic><sub>3</sub></td><td char=".">0.5</td><td char=".">0.0212</td><td char=".">0.2024</td><td char=".">0.2034</td><td char=".">94.8</td><td char=".">0.0051</td><td char=".">0.1236</td><td char=".">0.1195</td><td char=".">94.8</td></tr><tr><td /><td><italic>β</italic><sub>4</sub></td><td char=".">−0.5</td><td char=".">−0.0122</td><td char=".">0.1935</td><td char=".">0.2030</td><td char=".">96.2</td><td char=".">−0.0121</td><td char=".">0.1198</td><td char=".">0.1193</td><td char=".">94.4</td></tr><tr><td /><td><italic>β</italic><sub>5</sub></td><td char=".">0.5</td><td char=".">0.0180</td><td char=".">0.1948</td><td char=".">0.2032</td><td char=".">95.6</td><td char=".">0.0058</td><td char=".">0.1187</td><td char=".">0.1196</td><td char=".">95.5</td></tr><tr><td /><td><italic>γ</italic></td><td char=".">0.5</td><td char=".">0.0332</td><td char=".">0.1795</td><td char=".">0.1886</td><td char=".">96.9</td><td char=".">0.0145</td><td char=".">0.1088</td><td char=".">0.1077</td><td char=".">95.7</td></tr><tr><td><italic>Y</italic><sub>1</sub></td><td><p><graphic href="hsem_a_2368023_ilm0263.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>1</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0058</td><td char=".">0.0666</td><td char=".">0.0691</td><td char=".">95.6</td><td char=".">−0.0014</td><td char=".">0.0420</td><td char=".">0.0435</td><td char=".">96.4</td></tr><tr><td><italic>Y</italic><sub>2</sub></td><td><italic>ψ</italic><sub>2</sub></td><td char=".">1</td><td char=".">0.0069</td><td char=".">0.1100</td><td char=".">0.1184</td><td char=".">96.6</td><td char=".">0.0043</td><td char=".">0.0674</td><td char=".">0.0699</td><td char=".">95.5</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0264.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>2</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0076</td><td char=".">0.0674</td><td char=".">0.0686</td><td char=".">95.5</td><td char=".">−0.0012</td><td char=".">0.0413</td><td char=".">0.0434</td><td char=".">96.4</td></tr><tr><td><italic>Y</italic><sub>3</sub></td><td><italic>ψ</italic><sub>3</sub></td><td char=".">1</td><td char=".">0.0039</td><td char=".">0.1052</td><td char=".">0.1186</td><td char=".">96.7</td><td char=".">0.0059</td><td char=".">0.0665</td><td char=".">0.0700</td><td char=".">96.3</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0265.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>3</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0002</td><td char=".">0.0677</td><td char=".">0.0693</td><td char=".">96.3</td><td char=".">−0.0032</td><td char=".">0.0404</td><td char=".">0.0435</td><td char=".">96.3</td></tr><tr><td><italic>Y</italic><sub>4</sub></td><td><italic>ψ</italic><sub>4</sub></td><td char=".">1</td><td char=".">0.0028</td><td char=".">0.1061</td><td char=".">0.1181</td><td char=".">97.0</td><td char=".">0.0019</td><td char=".">0.0674</td><td char=".">0.0700</td><td char=".">96.0</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0266.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>4</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0017</td><td char=".">0.0677</td><td char=".">0.0689</td><td char=".">96.0</td><td char=".">0.0014</td><td char=".">0.0406</td><td char=".">0.0436</td><td char=".">96.6</td></tr><tr><td><italic>Y</italic><sub>5</sub></td><td><p><graphic href="hsem_a_2368023_ilm0267.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>5</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0267</td><td char=".">0.1441</td><td char=".">0.1543</td><td char=".">95.7</td><td char=".">−0.0070</td><td char=".">0.0897</td><td char=".">0.0985</td><td char=".">97.0</td></tr><tr><td><italic>Y</italic><sub>6</sub></td><td><italic>ψ</italic><sub>6</sub></td><td char=".">1</td><td char=".">0.0116</td><td char=".">0.2276</td><td char=".">0.2322</td><td char=".">94.1</td><td char=".">0.0072</td><td char=".">0.1442</td><td char=".">0.1570</td><td char=".">96.5</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0268.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>6</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0087</td><td char=".">0.1393</td><td char=".">0.1408</td><td char=".">95.2</td><td char=".">−0.0043</td><td char=".">0.0920</td><td char=".">0.0984</td><td char=".">96.1</td></tr><tr><td><italic>Y</italic><sub>7</sub></td><td><p><graphic href="hsem_a_2368023_ilm0269.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>7</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0062</td><td char=".">0.0752</td><td char=".">0.0746</td><td char=".">95.9</td><td char=".">−0.0003</td><td char=".">0.0448</td><td char=".">0.0469</td><td char=".">96.2</td></tr><tr><td><italic>Y</italic><sub>8</sub></td><td><italic>ψ</italic><sub>8</sub></td><td char=".">1</td><td char=".">0.0117</td><td char=".">0.1419</td><td char=".">0.1489</td><td char=".">96.5</td><td char=".">0.0104</td><td char=".">0.0819</td><td char=".">0.0871</td><td char=".">96.0</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0270.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>8</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0047</td><td char=".">0.0712</td><td char=".">0.0750</td><td char=".">96.6</td><td char=".">−0.0020</td><td char=".">0.0465</td><td char=".">0.0471</td><td char=".">95.2</td></tr><tr><td><italic>Y</italic><sub>9</sub></td><td><italic>ψ</italic><sub>9</sub></td><td char=".">1</td><td char=".">0.0060</td><td char=".">0.1369</td><td char=".">0.1480</td><td char=".">95.8</td><td char=".">0.0110</td><td char=".">0.0834</td><td char=".">0.0872</td><td char=".">95.9</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0271.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>9</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0027</td><td char=".">0.0718</td><td char=".">0.0749</td><td char=".">97.0</td><td char=".">−0.0020</td><td char=".">0.0462</td><td char=".">0.0472</td><td char=".">95.6</td></tr><tr><td><italic>η</italic><sub>1</sub></td><td><italic>ν</italic><sub>1</sub></td><td char=".">0.5</td><td char=".">0.0069</td><td char=".">0.1155</td><td char=".">0.1219</td><td char=".">96.7</td><td char=".">0.0041</td><td char=".">0.0703</td><td char=".">0.0713</td><td char=".">94.7</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0272.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub></mrow><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0003</td><td char=".">0.0954</td><td char=".">0.0978</td><td char=".">94.7</td><td char=".">−0.0044</td><td char=".">0.0599</td><td char=".">0.0604</td><td char=".">96.0</td></tr><tr><td><italic>η</italic><sub>2</sub></td><td><italic>ν</italic><sub>2</sub></td><td char=".">−0.5</td><td char=".">−0.0117</td><td char=".">0.1221</td><td char=".">0.1405</td><td char=".">96.8</td><td char=".">−0.0052</td><td char=".">0.0793</td><td char=".">0.0826</td><td char=".">95.6</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0273.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0194</td><td char=".">0.1440</td><td char=".">0.1631</td><td char=".">96.2</td><td char=".">0.0055</td><td char=".">0.0909</td><td char=".">0.0962</td><td char=".">97.2</td></tr><tr><td><italic>η</italic><sub>3</sub></td><td><p><graphic href="hsem_a_2368023_ilm0274.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub></mrow><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0033</td><td char=".">0.1062</td><td char=".">0.1070</td><td char=".">94.2</td><td char=".">−0.0032</td><td char=".">0.0634</td><td char=".">0.0661</td><td char=".">96.6</td></tr></tbody></table> </ephtml> </p> <p>Table 2. Simulation results for the SEM with the PO submodel (<emph>r</emph> = 1), including the estimation bias (bias), the sample standard error (SSE) of the 1000 estimates, the average of the 1000 standard error estimates (SEE), and the 95% coverage probability (CP). Each row corresponds to the parameter in the model or distribution of "resp".</p> <p> <ephtml> <table><thead><tr><td /><td /><td /><td><italic>n</italic> = 200</td><td><italic>n</italic> = 500</td></tr><tr><td>Resp</td><td>Para.</td><td>True</td><td>Bias</td><td>SSE</td><td>SEE</td><td>CP</td><td>Bias</td><td>SSE</td><td>SEE</td><td>CP</td></tr></thead><tbody valign="top"><tr><td><italic>T</italic></td><td><italic>β</italic><sub>1</sub></td><td char=".">0.5</td><td char=".">0.0217</td><td char=".">0.1587</td><td char=".">0.1575</td><td char=".">95.9</td><td char=".">0.0115</td><td char=".">0.0941</td><td char=".">0.0933</td><td char=".">94.1</td></tr><tr><td /><td><italic>β</italic><sub>2</sub></td><td char=".">−0.5</td><td char=".">−0.0114</td><td char=".">0.1550</td><td char=".">0.1572</td><td char=".">95.5</td><td char=".">−0.0047</td><td char=".">0.0910</td><td char=".">0.0930</td><td char=".">95.4</td></tr><tr><td /><td><italic>β</italic><sub>3</sub></td><td char=".">0.5</td><td char=".">0.0251</td><td char=".">0.3000</td><td char=".">0.2970</td><td char=".">95.3</td><td char=".">0.0063</td><td char=".">0.1833</td><td char=".">0.1738</td><td char=".">94.0</td></tr><tr><td /><td><italic>β</italic><sub>4</sub></td><td char=".">−0.5</td><td char=".">−0.0130</td><td char=".">0.2864</td><td char=".">0.2965</td><td char=".">95.5</td><td char=".">−0.0074</td><td char=".">0.1780</td><td char=".">0.1741</td><td char=".">94.4</td></tr><tr><td /><td><italic>β</italic><sub>5</sub></td><td char=".">0.5</td><td char=".">0.0184</td><td char=".">0.2841</td><td char=".">0.2965</td><td char=".">96.2</td><td char=".">0.0080</td><td char=".">0.1763</td><td char=".">0.1739</td><td char=".">94.8</td></tr><tr><td /><td><italic>γ</italic></td><td char=".">0.5</td><td char=".">0.0282</td><td char=".">0.2572</td><td char=".">0.2658</td><td char=".">96.2</td><td char=".">0.0168</td><td char=".">0.1569</td><td char=".">0.1539</td><td char=".">94.3</td></tr><tr><td><italic>Y</italic><sub>1</sub></td><td><p><graphic href="hsem_a_2368023_ilm0275.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>1</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0058</td><td char=".">0.0666</td><td char=".">0.0691</td><td char=".">95.7</td><td char=".">−0.0014</td><td char=".">0.0420</td><td char=".">0.0435</td><td char=".">96.5</td></tr><tr><td><italic>Y</italic><sub>2</sub></td><td><italic>ψ</italic><sub>2</sub></td><td char=".">1</td><td char=".">0.0069</td><td char=".">0.1099</td><td char=".">0.1184</td><td char=".">96.5</td><td char=".">0.0043</td><td char=".">0.0675</td><td char=".">0.0699</td><td char=".">95.6</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0276.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>2</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0076</td><td char=".">0.0675</td><td char=".">0.0686</td><td char=".">95.5</td><td char=".">−0.0012</td><td char=".">0.0413</td><td char=".">0.0434</td><td char=".">96.4</td></tr><tr><td><italic>Y</italic><sub>3</sub></td><td><italic>ψ</italic><sub>3</sub></td><td char=".">1</td><td char=".">0.0039</td><td char=".">0.1052</td><td char=".">0.1186</td><td char=".">96.7</td><td char=".">0.0059</td><td char=".">0.0665</td><td char=".">0.0700</td><td char=".">96.3</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0277.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>3</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0002</td><td char=".">0.0677</td><td char=".">0.0693</td><td char=".">96.3</td><td char=".">−0.0032</td><td char=".">0.0404</td><td char=".">0.0435</td><td char=".">96.3</td></tr><tr><td><italic>Y</italic><sub>4</sub></td><td><italic>ψ</italic><sub>4</sub></td><td char=".">1</td><td char=".">0.0028</td><td char=".">0.1061</td><td char=".">0.1181</td><td char=".">97.2</td><td char=".">0.0020</td><td char=".">0.0674</td><td char=".">0.0700</td><td char=".">96.0</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0278.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>4</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0017</td><td char=".">0.0677</td><td char=".">0.0689</td><td char=".">96.0</td><td char=".">0.0014</td><td char=".">0.0406</td><td char=".">0.0436</td><td char=".">96.5</td></tr><tr><td><italic>Y</italic><sub>5</sub></td><td><p><graphic href="hsem_a_2368023_ilm0279.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>5</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0269</td><td char=".">0.1458</td><td char=".">0.1548</td><td char=".">95.6</td><td char=".">−0.0069</td><td char=".">0.0903</td><td char=".">0.0987</td><td char=".">97.1</td></tr><tr><td><italic>Y</italic><sub>6</sub></td><td><italic>ψ</italic><sub>6</sub></td><td char=".">1</td><td char=".">0.0118</td><td char=".">0.2295</td><td char=".">0.2395</td><td char=".">94.3</td><td char=".">0.0077</td><td char=".">0.1452</td><td char=".">0.1574</td><td char=".">96.5</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0280.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>6</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0091</td><td char=".">0.1410</td><td char=".">0.1413</td><td char=".">94.9</td><td char=".">−0.0046</td><td char=".">0.0924</td><td char=".">0.0986</td><td char=".">96.2</td></tr><tr><td><italic>Y</italic><sub>7</sub></td><td><p><graphic href="hsem_a_2368023_ilm0281.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>7</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0067</td><td char=".">0.0760</td><td char=".">0.0752</td><td char=".">95.6</td><td char=".">−0.0002</td><td char=".">0.0452</td><td char=".">0.0473</td><td char=".">96.3</td></tr><tr><td><italic>Y</italic><sub>8</sub></td><td><italic>ψ</italic><sub>8</sub></td><td char=".">1</td><td char=".">0.0111</td><td char=".">0.1427</td><td char=".">0.1500</td><td char=".">96.3</td><td char=".">0.0108</td><td char=".">0.0825</td><td char=".">0.0878</td><td char=".">96.7</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0282.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>8</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0044</td><td char=".">0.0717</td><td char=".">0.0757</td><td char=".">96.7</td><td char=".">−0.0023</td><td char=".">0.0470</td><td char=".">0.0475</td><td char=".">95.4</td></tr><tr><td><italic>Y</italic><sub>9</sub></td><td><italic>ψ</italic><sub>9</sub></td><td char=".">1</td><td char=".">0.0053</td><td char=".">0.1384</td><td char=".">0.1490</td><td char=".">95.2</td><td char=".">0.0109</td><td char=".">0.0840</td><td char=".">0.0878</td><td char=".">95.7</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0283.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mn>9</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">−0.0024</td><td char=".">0.0723</td><td char=".">0.0755</td><td char=".">97.1</td><td char=".">−0.0018</td><td char=".">0.0468</td><td char=".">0.0475</td><td char=".">95.3</td></tr><tr><td><italic>η</italic><sub>1</sub></td><td><italic>ν</italic><sub>1</sub></td><td char=".">0.5</td><td char=".">0.0069</td><td char=".">0.1159</td><td char=".">0.1223</td><td char=".">96.7</td><td char=".">0.0040</td><td char=".">0.0704</td><td char=".">0.0715</td><td char=".">94.9</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0284.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub></mrow><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0003</td><td char=".">0.0954</td><td char=".">0.0978</td><td char=".">94.6</td><td char=".">−0.0044</td><td char=".">0.0599</td><td char=".">0.0604</td><td char=".">96.1</td></tr><tr><td><italic>η</italic><sub>2</sub></td><td><italic>ν</italic><sub>2</sub></td><td char=".">−0.5</td><td char=".">−0.0115</td><td char=".">0.1225</td><td char=".">0.1408</td><td char=".">97.0</td><td char=".">−0.0050</td><td char=".">0.0795</td><td char=".">0.0828</td><td char=".">95.6</td></tr><tr><td /><td><p><graphic href="hsem_a_2368023_ilm0285.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0198</td><td char=".">0.1448</td><td char=".">0.1635</td><td char=".">96.0</td><td char=".">0.0055</td><td char=".">0.0915</td><td char=".">0.0963</td><td char=".">97.2</td></tr><tr><td><italic>η</italic><sub>3</sub></td><td><p><graphic href="hsem_a_2368023_ilm0286.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi><mn /></mrow><mrow><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub></mrow><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.5</td><td char=".">0.0038</td><td char=".">0.1069</td><td char=".">0.1075</td><td char=".">94.2</td><td char=".">−0.0032</td><td char=".">0.0637</td><td char=".">0.0664</td><td char=".">96.8</td></tr></tbody></table> </ephtml> </p> <p>We also considered the setting where some manifest variables were generated from non-normal distribution, such as the binary distribution. The detailed simulation specifications are presented in Appendix D of the Supplementary Material. Results in Tables S1 and S2 of the Supplementary Material show similar findings.</p> <hd id="AN0182192620-11">6. An Application</hd> <p>The ADNI study was initiated in 2004 and recruited over 2,000 people across North America. The participants were further stratified into three categories according to their cognitive conditions: cognitive normal, mild cognitive impairment (MCI) and AD. In our analysis below, as in Li et al. ([<reflink idref="bib18" id="ref73">18</reflink>]) and others, we focused on the participants in the MCI group who had already experienced cognitive impairment at the study recruitment. The failure time of interest <emph>T</emph> is defined as the time from the recruitment to the AD conversion, which was determined intermittently at the periodic examination times and cannot be monitored precisely. Thus, the failure time <emph>T</emph> suffers from interval censoring.</p> <p>In this study, the potential risk factors considered for AD incorporate a genetic covariate APOE<emph>ϵ</emph>4 (the number of APOE<emph>ϵ</emph>4 alleles coded as 0, 1, and 2) and four demographic covariates measured at the baseline visit: Age (given in years), Gender (1 for male and 0 for female), Education (years of education) and Marry (marital status, 1 for married and 0 otherwise). In addition to the above covariates, we used the proposed SEM to characterize the degree of cognitive impairment of each individual at the recruitment, a potential risk factor for AD development (Desgranges et al., [<reflink idref="bib5" id="ref74">5</reflink>]; Drijgers et al., [<reflink idref="bib6" id="ref75">6</reflink>]). As discussed in Section 1, cognitive impairment is a latent variable measured by multiple test scores related to an individual's cognitive ability and simultaneously reflected by memory status (Memory) and executive function (Executive function). In this ADNI study, the AD Assessment Scale Cognitive Subscales (ADAS) 11 and 13 and the Mini-Mental State Examination (MMSE) jointly characterize an individual's cognitive ability. The test scores measured by ADAS 11, ADAS 13, and MMSE were denoted as ADAS11, ADAS13, and MMSE, respectively. Higher values of ADAS11 and ADAS13 and lower values of MMSE correspond to worse cognitive ability.</p> <p>As suggested by Park et al. ([<reflink idref="bib25" id="ref76">25</reflink>]) and others, an individual's memory and executive function are also latent variables integrated by some measured surrogates. In particular, Memory can be represented by four scores obtained from the Rey Auditory Verbal Learning Test (RAVLT): the immediate recall (RAVLT.i), learning ability (RAVLT.l), 30-min delayed recall (RAVLT.d), and 30-min delayed recognition (DelayRec). The Executive function can be summarized by two scores TrailA and TrailB obtained from the Trail Making Test Part A and Part B, respectively. Higher values of RAVLT.i, RAVLT.l, RAVLT.d, and DelayRec correspond to better performance in immediate recall, learning ability, 30-min delayed recall, and 30-min delayed recognition tests, respectively. Meanwhile, higher values of TrailA and TrailB indicate longer processing times to complete the tasks in Part A and Part B, representing a worse executive function. By excluding a small proportion of individuals with missing covariates, the final sample size is <emph>n</emph> = 361, including 31 left-censored and 200 right-censored observations. In the subsequent analysis, the continuous variables were standardized to have a mean of 0 and a variance of 1.</p> <p>Figure 3 depicts the path diagram of the proposed SEM, including three latent factors enclosed by ellipses, nine manifest variables enclosed by rectangles, and five other observed covariates enclosed by hexagons. Specifically, we set</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">η</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup><mo>=</mo></mrow></math> </ephtml> (Memory, Executivefunction, Cognitiveimpairment)<sups>T</sups> in the structural model (<reflink idref="bib1" id="ref77">1</reflink>),</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">Y</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mn>9</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> = (RAVLT.i, RAVLT.l, RAVLT.d, DelayRec, TrailA, TrailB, ADAS11, ADAS13, MMSE)<sups>T</sups> in the measurement model (<reflink idref="bib2" id="ref78">2</reflink>), and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">X</mi><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mn>5</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">T</mi></msup></mrow></math> </ephtml> = (Age, Gender, Education, Marry, APOE</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow></math> </ephtml> 4)<sups>T</sups> in the transformation model (<reflink idref="bib3" id="ref79">3</reflink>). To ensure model identifiability, the regression parameters of <emph>η</emph><subs>1</subs> in the model of <emph>Y</emph><subs>1</subs>, <emph>η</emph><subs>2</subs> in the model of <emph>Y</emph><subs>5</subs>, and <emph>η</emph><subs>3</subs> in the model of <emph>Y</emph><subs>7</subs> are fixed to one.</p> <p>To examine the effects of both observed covariates and the degree of latent cognitive impairment on the hazard of developing AD, we used the factor-augmented transformation model (<reflink idref="bib3" id="ref80">3</reflink>) with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo /><mtext mathvariant="normal">log</mtext><mo /><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mtext mathvariant="italic">rx</mtext><mo stretchy="false">)</mo><mo>/</mo><mi>r</mi><mtext /><mo stretchy="false">(</mo><mi>r</mi><mo>≥</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> Model (<reflink idref="bib3" id="ref81">3</reflink>) reduces to the factor-augmented PH and PO models when setting <emph>r</emph> = 0 and <emph>r</emph> = 1, respectively. We let <emph>r</emph> vary from 0 to 2 with an increment of 0.1 and selected the optimal <emph>r</emph> value based on the maximum likelihood principle. Notably, the maximum likelihood principle is equivalent to the well-known model selection criteria BIC and AIC since the number of unknown parameters is the same under different <emph>r</emph> values. Figure 4 shows that the log-likelihood reached the maximum at <emph>r</emph> = 0.9, revealing that the model with <emph>r</emph> = 0.9 is optimal among all models considered.</p> <p>Figure 3 also reports the estimation results with the model with <emph>r</emph> = 0.9, including the regression parameter estimates and their standard error estimates. One can draw the following conclusions from Figure 3. First, all the estimated factor loadings of Memory are positive and significant, implying that high values of RAVLT.i, RAVLT.l, RAVLT.d, and DelayRec correspond to a high score of Memory and good memory status. Similarly, the estimated factor loading between TrailB and Executive function is also significantly positive. Thus, high values of TrailA and TrailB represent a high score of Executive function and a poor executive function. Second, based on the estimated factor loadings of cognitive impairment, high values of ADAS11 and ADAS13 and low value of MMSE correspond to a high score of Cognitive impairment, a sign of poor cognitive ability. Moreover, Cognitive impairment is negatively associated with Memory but positively associated with Excutive function. That is, cognitive impairment induces memory loss and reduced executive function. Third, based on the estimated regression parameters in the survival model, Cognitive impairment exerts a significantly positive effect on the AD hazard, suggesting that poor cognitive ability increases the risk of developing AD. APOE<emph>ϵ</emph>4 exhibits a significantly positive effect, indicating that carrying APOE<emph>ϵ</emph>4 alleles increases the AD hazard and that carrying two alleles has a higher risk of developing AD than carrying only one allele. For comparison, we also analyzed the data using the PH and PO submodels. Figures 5 and 6 show the estimation results and give the same conclusions as the optimal model.</p> <p>Graph: Figure 4. Calculated log-likelihood with respect to r.</p> <p>DIAGRAM: Figure 5. Path diagram of the used SEM with analysis results for the ADNI data under the PH submodel. The results include the parameter estimates and their standard error estimates given in brackets.</p> <p>DIAGRAM: Figure 6. Path diagram of the used SEM with analysis results for the ADNI data under the PO submodel. The results include the parameter estimates and their standard error estimates given in brackets.</p> <p>In conclusion, through the above ADNI data analysis, the proposed SEM was found to have critical practical utility in depicting the complex relationships among multiple variables, both latent and observed. In particular, it can comprise a joint analysis of the interrelationships among latent factors, such as Cognitive impairment, Memory, Executive function, and AD development.</p> <hd id="AN0182192620-12">7. Discussion</hd> <p>This study proposed a general semiparametric SEM framework that can simultaneously formulate the latent and observed covariate effects on the interval-censored outcome. Semiparametric factor-augmented transformation models include some popular regression models with latent variables as special cases and thus have flexible modeling capability. The proposed EM algorithm was computationally stable and reliable, given the satisfactory performance in simulation studies. The asymptotic properties of the proposed estimators further supported our findings in finite samples. An application to the ADNI study demonstrated the practical utility of the proposed methodology.</p> <p>In the proposed method, we used the multivariate Gaussian–Hermite quadrature to calculate the integral of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> with no explicit form. To speed up the computation, we implemented the proposed algorithm under the Rcpp environment in R. This technique makes the proposed method quite feasible when the dimension of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is small (e.g., 2 or 3). For example, in the simulations, the proposed algorithm took less than one minute to obtain the parameter estimate and its standard error based on one simulated data set with <emph>n</emph> = 200. However, the proposed method could become computationally intensive when the dimension of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is large. Our experiences show that, with a 5-dimensional</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">η</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> the proposed method required approximately 12 h to perform estimation in a single replication with <emph>n</emph> = 200. Such computational burden may be acceptable in an application but poses a significant challenge when conducting simulations. Under such a situation, we suggest using parallel computing with many cores (e.g.,100) to facilitate efficient computation.</p> <p>This work could be generalized in several directions. First, in the ADNI data analysis, we used the maximum likelihood principle to select the optimal transformation function in the class of factor-augmented models (<reflink idref="bib3" id="ref82">3</reflink>). However, developing formal diagnostic procedures to assess the fitness of the transformation function and other model assumptions to the data is an important topic and is worth further investigation. Second, we made a non-informative censoring assumption in the proposed SEM framework (Ma et al., [<reflink idref="bib21" id="ref83">21</reflink>]), which assumes that the failure and observation times are conditionally independent given the latent and observed covariates. However, this assumption may not hold in some specific fields. We can extend our work to handle informative interval-censored data by positing additional regression models for the observation times. Finally, generalizing the proposed method to other regression models, such as the additive hazards and accelerated failure time models (Chiou et al., [<reflink idref="bib3" id="ref84">3</reflink>]; Jin et al., [<reflink idref="bib12" id="ref85">12</reflink>]; Lin et al., [<reflink idref="bib19" id="ref86">19</reflink>]), deserves future study.</p> <hd id="AN0182192620-13">Acknowledgment</hd> <p>We are grateful to the Editor and two anonymous reviewers for their helpful suggestions and comments that greatly improved this article.</p> <hd id="AN0182192620-14">Data Availability Statement</hd> <p>The data used in the paper are not publicly available but can be requested from https://adni.loni.usc.edu/.</p> <p>Correction Statement</p> <p>This article has been corrected with minor changes. These changes do not impact the academic content of the article.</p> <ref id="AN0182192620-15"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref15" type="bt">1</bibl> <bibtext> Supplemental data for this article can be accessed online at https://doi.org/10.1080/10705511.2024.2368023.</bibtext> </blist> </ref> <ref id="AN0182192620-16"> <title> References </title> <blist> <bibtext> Asparouhov, T., Masyn, K. E., & Muthén, B. (2006). Continuous time survival in latent variable models. Proceedings of the Joint Statistical Meeting in Seattle, August 2006. ASA Section on Biometrics, 180 – 187.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref1" type="bt">2</bibl> <bibtext> Bollen, K. A. (1989). Structural equations with latent variables. Wiley.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref41" type="bt">3</bibl> <bibtext> Chiou, S. H., Kang, S., & Yan, J. (2015). Semiparametric accelerated failure time modeling for clustered failure times from stratified sampling. 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  Data: <searchLink fieldCode="AR" term="%22Hongxi+Li%22">Hongxi Li</searchLink><br /><searchLink fieldCode="AR" term="%22Shuwei+Li%22">Shuwei Li</searchLink><br /><searchLink fieldCode="AR" term="%22Liuquan+Sun%22">Liuquan Sun</searchLink><br /><searchLink fieldCode="AR" term="%22Xinyuan+Song%22">Xinyuan Song</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-4877-3200">0000-0002-4877-3200</externalLink>)
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  Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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  Data: Structural equation models offer a valuable tool for delineating the complicated interrelationships among multiple variables, including observed and latent variables. Over the last few decades, structural equation models have successfully analyzed complete and right-censored survival data, exemplified by wide applications in psychological, social, or genomic studies. However, the existing methodology for structural equation modeling is not concerned with interval-censored data, a type of coarse survival data arising typically from periodic examinations for the occurrence of asymptomatic disease. The present study aims to fill this gap and provide a flexible semiparametric structural equation modeling framework. A general class of factor-augmented transformation models is proposed to model the interval-censored outcome of interest in the presence of latent risk factors. An expectation-maximization algorithm is subtly designed to conduct the nonparametric maximum likelihood estimation. Furthermore, the asymptotic properties of the proposed estimators are established by leveraging the empirical process theory. The numerical results obtained from extensive simulations and an application to the Alzheimer's disease data set demonstrate the proposed method's empirical performance and practical utility.
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