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Exploring and Correcting the Bias in the Estimation of the Gini Measure of Inequality

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Title: Exploring and Correcting the Bias in the Estimation of the Gini Measure of Inequality
Language: English
Authors: Juan F. Muñoz (ORCID 0000-0001-7427-6630), Pablo J. Moya-Fernández (ORCID 0000-0003-0980-3849), Encarnación Álvarez-Verdejo (ORCID 0000-0002-0473-6037)
Source: Sociological Methods & Research. 2025 54(1):237-274.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 38
Publication Date: 2025
Document Type: Journal Articles
Reports - Evaluative
Descriptors: Bias, Educational Indicators, Equal Education, Monte Carlo Methods, Sampling, Statistical Inference, Income
DOI: 10.1177/00491241231176847
ISSN: 0049-1241
1552-8294
Abstract: The Gini index is probably the most commonly used indicator to measure inequality. For continuous distributions, the Gini index can be computed using several equivalent formulations. However, this is not the case with discrete distributions, where controversy remains regarding the expression to be used to estimate the Gini index. We attempt to bring a better understanding of the underlying problem by regrouping and classifying the most common estimators of the Gini index proposed in both infinite and finite populations, and focusing on the biases. We use Monte Carlo simulation studies to analyse the bias of the various estimators under a wide range of scenarios. Extremely large biases are observed in heavy-tailed distributions with high Gini indices, and bias corrections are recommended in this situation. We propose the use of some (new and traditional) bootstrap-based and jackknife-based strategies to mitigate this bias problem. Results are based on continuous distributions often used in the modelling of income distributions. We describe a simulation-based criterion for deciding when to use bias corrections. Various real data sets are used to illustrate the practical application of the suggested bias corrected procedures.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1457727
Database: ERIC
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  Value: <anid>AN0182244234;som01feb.25;2025Jan17.04:27;v2.2.500</anid> <title id="AN0182244234-1">Exploring and Correcting the Bias in the Estimation of the Gini Measure of Inequality </title> <p>The Gini index is probably the most commonly used indicator to measure inequality. For continuous distributions, the Gini index can be computed using several equivalent formulations. However, this is not the case with discrete distributions, where controversy remains regarding the expression to be used to estimate the Gini index. We attempt to bring a better understanding of the underlying problem by regrouping and classifying the most common estimators of the Gini index proposed in both infinite and finite populations, and focusing on the biases. We use Monte Carlo simulation studies to analyse the bias of the various estimators under a wide range of scenarios. Extremely large biases are observed in heavy-tailed distributions with high Gini indices, and bias corrections are recommended in this situation. We propose the use of some (new and traditional) bootstrap-based and jackknife-based strategies to mitigate this bias problem. Results are based on continuous distributions often used in the modelling of income distributions. We describe a simulation-based criterion for deciding when to use bias corrections. Various real data sets are used to illustrate the practical application of the suggested bias corrected procedures.</p> <p>Keywords: Gini index; income distribution; bootstrap; jackknife; survey sampling</p> <hd id="AN0182244234-2">Introduction</hd> <p>[<reflink idref="bib33" id="ref1">33</reflink>] and [<reflink idref="bib22" id="ref2">22</reflink>] were the first to develop measures of inequality. More than a century after these contributions, inequality analysis remains an active and essential topic in numerous fields. The Gini index, also often referred to as the Gini coefficient, is probably the most commonly used to measure inequality. This indicator ranges between 0 and 1, where 0 indicates perfect equality, and 1 the opposite. Inequality is of special interest in economic studies ([<reflink idref="bib42" id="ref3">42</reflink>]; [<reflink idref="bib50" id="ref4">50</reflink>]), and the Gini index is used especially to measure income inequality. Many studies indicate that income inequality has been increasing overall in recent years ([<reflink idref="bib8" id="ref5">8</reflink>]), and marked differences can be observed across various countries. For instance, results from the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>U</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>I</mi><mi>L</mi><mi>C</mi></math> </ephtml> (European Union Statistics on Income and Living Conditions) survey show that Slovakia has the smallest Gini index estimate ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.21</mn></math> </ephtml> ), while Turkey ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.43</mn></math> </ephtml> ) has the highest of all the countries in the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>U</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>I</mi><mi>L</mi><mi>C</mi></math> </ephtml> . At a global level, the World Bank indicates that South Africa has the highest Gini index ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.63</mn></math> </ephtml> ). These results are associated with countries, but more extreme values are expected at regional level, in subpopulations, small areas, etc.</p> <p>The aforementioned differences across countries indicate that institutions and policies may have an important role to play in reducing inequality. In fact, reducing inequality is one of the 17 Sustainable Development Goals of the United Nations 2030 Agenda for Sustainable Development. The Gini index is a common statistical tool employed by the 2030 Agenda for measuring inequality (see [<reflink idref="bib49" id="ref6">49</reflink>]). However, it should be noted that in order to reduce inequality, it is crucial to be able to accurately measure this phenomenon without biases and/or errors. Indeed, [<reflink idref="bib18" id="ref7">18</reflink>] argued that a large bias is usually an undesirable aspect of an estimator's performance. The relevance of the Gini index has also been demonstrated by its use to describe inequality in many fields and among different socioeconomic groups, such as length of life or wellbeing ([<reflink idref="bib55" id="ref8">55</reflink>]), educational opportunity ([<reflink idref="bib9" id="ref9">9</reflink>]), housing prices ([<reflink idref="bib53" id="ref10">53</reflink>]), gender inequality ([<reflink idref="bib30" id="ref11">30</reflink>]; [<reflink idref="bib32" id="ref12">32</reflink>]), input and outcome inequality ([<reflink idref="bib27" id="ref13">27</reflink>]), and horizontal and vertical inequality ([<reflink idref="bib11" id="ref14">11</reflink>]).</p> <p>There is a formal theoretical definition of the Gini index for continuous distributions, and many equivalent formulations have been proposed in the literature. As discussed by [<reflink idref="bib14" id="ref15">14</reflink>], there is no disagreement about the definition of the Gini index for continuous distributions, since the various existing expressions provide the same outcome. However, for discrete distributions, many different formulations have been suggested in the extensive literature, and there has been notable controversy surrounding the appropriate version to use in this scenario (see also [<reflink idref="bib29" id="ref16">29</reflink>]). For discrete distributions, various expressions of the Gini index are plug-in formulations of theoretical definitions of the Gini index for continuous distributions. Throughout this article, we refer to this value derived from a continuous distribution as the true value of the Gini index, and formulations of the Gini index for discrete distributions are referred to as empirical versions or estimators of the true Gini index. A highly debated topic in the literature is whether or not to use a specific bias corrected estimator, which is denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> in this article. [<reflink idref="bib26" id="ref17">26</reflink>], [<reflink idref="bib15" id="ref18">15</reflink>] and [<reflink idref="bib14" id="ref19">14</reflink>] provide some arguments in favour of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> .</p> <p>Statistical techniques can be based on infinite or finite populations. Classical statistical theory assumes that sampled units are independently selected from an infinite population, whereas survey sampling theory (see [<reflink idref="bib47" id="ref20">47</reflink>]) considers that samples are selected from a finite population. Survey sampling has specific features, and this implies that statistical techniques designed for infinite populations must be modified so that they can be used for finite populations. For instance, the usual assumption of independence is not satisfied in finite populations when samples are selected without replacement. Note also that the use of continuous probabilistic distributions to model income and wealth distributions is common practice in many real-world applications. For instance, the Dagum, Pareto, Weibull and Gamma distributions are used, respectively, by [<reflink idref="bib40" id="ref21">40</reflink>], [<reflink idref="bib2" id="ref22">2</reflink>], [<reflink idref="bib3" id="ref23">3</reflink>] and [<reflink idref="bib46" id="ref24">46</reflink>]. The Lognormal distribution is often used to model household income in many countries (see [<reflink idref="bib12" id="ref25">12</reflink>]).</p> <p>This article describes, in the second section, the most common formulations for calculating the Gini index from both discrete and continuous distributions, and in scenarios of infinite and finite populations. The first aim of this article is to regroup and classify existing empirical versions of the Gini index, and provide a better overview of the problem of estimating this parameter. The second aim is to analyse, in the third section, the biases of different versions of the Gini index. For this purpose, we consider a variety of Gini indices and various probabilistic distributions commonly used to model income distributions. Our results reveal that extremely large biases may appear, especially for heavy-tailed distributions and large Gini indices, and bias correction procedures are recommended in this situation. As expected, the bias problem is more serious in small samples, as is the case of rural studies ([<reflink idref="bib54" id="ref26">54</reflink>]), small areas ([<reflink idref="bib20" id="ref27">20</reflink>]), subpopulations (Särndal, Swensson, and Wretman 2003:386), etc. The third contribution is to describe, in the fourth section, bias correction procedures that may reduce the aforementioned large biases. Bootstrap and jackknife methods are considered, and a novel empirical bootstrap is also adapted to the problem of estimating the Gini index. In the fifth section, the bias correction procedures are analysed using Monte Carlo simulation studies. The sixth section describes a simulation-based criterion for deciding when to use bias correction procedures, which are then illustrated, in the seventh section, by application to various real data sets. Finally, a brief discussion is presented in the eighth section. The Supplemental material contains: (i) the selected parameters of the analysed probabilistic distributions; (ii) results from simulation studies based on large samples ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>500</mn></math> </ephtml> ); (iii) description of the bias functions suggested in the fourth section, and information on their percentages of use and (iv) efficiency and bias ratios of estimators of the Gini index, which are explored in the fifth and the sixth sections.</p> <hd id="AN0182244234-3">The Gini Index</hd> <p></p> <hd id="AN0182244234-4">Definition</hd> <p>We assume that inequality is analysed using a variable of interest <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , which is a non-negative continuous random variable. A popular formulation of the Gini index is defined in terms of the average absolute difference between each possible pair of individuals ([<reflink idref="bib43" id="ref28">43</reflink>]), that is, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><msub><mi>μ</mi><mi>Y</mi></msub></mrow></mfrac></mrow><munderover><mrow><mo largeop="false">∫</mo></mrow><mn>0</mn><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mspace width="0.2em" /><munderover><mrow><mo largeop="false">∫</mo></mrow><mn>0</mn><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mspace width="0.2em" /><mo fence="false" stretchy="false">|</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>d</mi><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>Y</mi></msub><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">]</mo><mo>=</mo><munderover><mrow><mo largeop="false">∫</mo></mrow><mn>0</mn><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mspace width="0.2em" /><mi>y</mi><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi><mo>=</mo><munderover><mrow><mo largeop="false">∫</mo></mrow><mn>0</mn><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mspace width="0.2em" /><mi>y</mi><mi>d</mi><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>is the mean of <emph>Y</emph>, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mrow><mi>Y</mi><mo>≤</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math> </ephtml> are, respectively, the distribution function and the probability density function of <emph>Y</emph>. A formulation of <emph>G</emph> based on the distribution function is ([<reflink idref="bib5" id="ref29">5</reflink>]; Qin, Rao, and Wu 2010): <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><msub><mi>μ</mi><mi>Y</mi></msub></mrow></mfrac></mrow><munderover><mrow><mo largeop="false">∫</mo></mrow><mn>0</mn><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mspace width="0.2em" /><mo fence="false" stretchy="false">{</mo><mn>2</mn><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo><mi>y</mi><mi>d</mi><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>[<reflink idref="bib1" id="ref30">1</reflink>] showed that the Gini index <emph>G</emph> can be computed as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>/</mo><msub><mi>μ</mi><mi>Y</mi></msub></math> </ephtml> times the covariance between <emph>Y</emph> and the distribution function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math> </ephtml> , that is, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>2</mn><mrow><msub><mi>μ</mi><mi>Y</mi></msub></mrow></mfrac></mrow><mi>c</mi><mi>o</mi><mi>v</mi><mo fence="false" stretchy="false">{</mo><mrow><mi>Y</mi><mo>,</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo fence="false" stretchy="false">}</mo><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>Finally, [<reflink idref="bib60" id="ref31">60</reflink>] and [<reflink idref="bib5" id="ref32">5</reflink>] consider the expression <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mn>1</mn><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>μ</mi><mi>Z</mi></msub></mrow><mrow><msub><mi>μ</mi><mi>Y</mi></msub></mrow></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>Z</mi></msub><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mrow><mo largeop="false">∫</mo></mrow><mn>0</mn><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><mspace width="0.2em" /><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo><mi>d</mi><mi>z</mi></math> </ephtml> is the expectation of the minimum <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mrow><mi mathvariant="normal">min</mi></mrow><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>Y</mi><mi>a</mi></msub><mo>,</mo><msub><mi>Y</mi><mi>b</mi></msub></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>a</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>b</mi></msub></math> </ephtml> are two independent random variables with the same distribution as <emph>Y</emph>. For continuous distributions, the Gini index can be defined in many other ways, as can be seen in [<reflink idref="bib60" id="ref33">60</reflink>], [<reflink idref="bib23" id="ref34">23</reflink>], etc. In practice, the value of <emph>G</emph> is estimated by means of a sample <emph>S</emph>, with size <emph>n</emph>, and which can be selected from either infinite or finite populations ([<reflink idref="bib29" id="ref35">29</reflink>]). The estimation of <emph>G</emph> under both scenarios is discussed in "Estimation" subsection.</p> <hd id="AN0182244234-5">Estimation</hd> <p>For infinite populations, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>Y</mi><mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>S</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> are considered as a sequence, with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> </ephtml> , of non-negative random variables with the same distribution as the variable of interest <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . The Gini index is estimated using an estimator of <emph>G</emph> based on the observations of individuals selected in the sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> , and which are denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>S</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> . Such estimators are usually defined as plug-in formulations derived from a theoretical definition of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> . This methodology may introduce a bias in comparison to the true parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> , especially for extreme values of the Gini index. As can be seen in the third section, a notable example is the plug-in expression of equation (<reflink idref="bib2" id="ref36">2</reflink>), which is defined as (see [<reflink idref="bib5" id="ref37">5</reflink>]; Qin, Rao, and Wu 2010): <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><mo stretchy="false">[</mo><mrow><mo fence="false" stretchy="false">{</mo><mrow><mn>2</mn><msub><mrow><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow><mo fence="false" stretchy="false">}</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><mo stretchy="false">]</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>2</mn><mrow><mi>n</mi><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>y</mi><mi>i</mi></msub><msub><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo>,</mo></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow><mo>=</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>y</mi><mi>i</mi></msub></math> </ephtml> is the sample mean, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><mi>δ</mi><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>≤</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> the sample (empirical) distribution function and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> the indicator variable that takes the value 1 if its argument is true and 0 otherwise. The classical empirical version of <emph>G</emph> ([<reflink idref="bib23" id="ref38">23</reflink>]) is the plug-in expression of equation (<reflink idref="bib1" id="ref39">1</reflink>), that is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>n</mi><mn>2</mn></msup><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mspace width=".1em" /><mi>j</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><mo fence="false" stretchy="false">|</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><msub><mi>y</mi><mi>j</mi></msub></mrow><mo fence="false" stretchy="false">|</mo><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>Note that many equivalent versions of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> have been suggested in the extensive literature on the Gini index. For instance, [<reflink idref="bib48" id="ref40">48</reflink>] proposed the popular formulation <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>1</mn></mrow></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>2</mn><mrow><msup><mi>n</mi><mn>2</mn></msup><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><mi>i</mi><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>n</mi></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>2</mn><mrow><msup><mi>n</mi><mn>2</mn></msup><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>r</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>n</mi></mfrac></mrow><mo>,</mo></mstyle></mstyle></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></math> </ephtml> are the values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>i</mi></msub></math> </ephtml> sorted in increasing order and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>i</mi></msub></math> </ephtml> the rank of unit <emph>i</emph> in the sample <emph>S</emph>. Similarly, the Gini index can be defined using the regression coefficient of an ordinary least squares regression (see [<reflink idref="bib38" id="ref41">38</reflink>]). This is the idea behind <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mn>2</mn><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></mfrac></mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>n</mi></mfrac></mrow><mo>,</mo></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>which assumes the regression model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mi>β</mi><mo>+</mo><msub><mi>u</mi><mi>i</mi></msub></math> </ephtml> , and where the heterocesdatic error <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>i</mi></msub></math> </ephtml> has variance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup><mo>/</mo><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></math> </ephtml> . The least squares estimator of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> is given by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></msub><mspace width="0.2em" /><mi>i</mi><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mrow><msub><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></msub><mspace width="0.2em" /><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mfrac></mrow><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>Finally, an equivalent version of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> is the empirical version of equation (<reflink idref="bib3" id="ref42">3</reflink>), that is, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>3</mn></mrow></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>2</mn><mrow><mi>n</mi><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><mi>c</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mrow><mi>i</mi><mo>,</mo><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mo stretchy="false">)</mo><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mrow><mi>i</mi><mo>,</mo><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><mi>i</mi><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow><mo>.</mo></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>The estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and its equivalent versions satisfy the symmetry axiom of [<reflink idref="bib48" id="ref43">48</reflink>], which establishes that an estimator of <emph>G</emph> based on a set of observations, say <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>S</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> , must coincide with the Gini index estimated by means of the same approach but using the sample <emph>S</emph> exactly replicated, that is, doubled in size (see [<reflink idref="bib14" id="ref44">14</reflink>]). Alternatively, the bias corrected estimator <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mi>n</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></mstyle></math> </ephtml></p> <p>Graph</p> <p>is often used instead of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> . Some equivalent expressions of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> are: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mn>1</mn></mrow></msubsup></mrow><mo>=</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>2</mn><mrow><mi>n</mi><mo stretchy="false">(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo stretchy="false">)</mo><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><mi>i</mi><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mo>;</mo></mstyle></mstyle></mrow></mtd></mtr><mtr><mtd /><mtd /></mtr><mtr><mtd><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mn>2</mn></mrow></msubsup></mrow><mo>=</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><msup><mrow><mrow><mo>(</mo><mrow><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mn>1</mn><mo>≤</mo><msub><mi>i</mi><mn>1</mn></msub><mo><</mo><msub><mi>i</mi><mn>2</mn></msub><mo>≤</mo><mi>n</mi></mrow></munder><mspace width="0.2em" /><mo fence="false" stretchy="false">|</mo><mrow><msub><mi>y</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow><mo fence="false" stretchy="false">|</mo><mo>;</mo></mstyle></mrow></mtd></mtr><mtr><mtd /><mtd /><mtd /></mtr><mtr><mtd><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mn>3</mn></mrow></msubsup></mrow><mo>=</mo><mrow><mn>1</mn><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow></mrow><mrow><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></mfrac></mrow><mo>.</mo></mstyle></mrow></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>[<reflink idref="bib26" id="ref45">26</reflink>] suggested the use of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mn>1</mn></mrow></msubsup></math> </ephtml> , [<reflink idref="bib56" id="ref46">56</reflink>] consider <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mn>2</mn></mrow></msubsup></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mn>3</mn></mrow></msubsup></math> </ephtml> is used by [<reflink idref="bib5" id="ref47">5</reflink>], where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow><mo>=</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>z</mi><mrow><mi>i</mi><mo>.</mo><mi>n</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mrow><mi>i</mi><mo>.</mo><mi>n</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mrow><mspace width=".1em" /><mi>j</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mspace width=".1em" /><mi>j</mi><mo>≠</mo><mi>i</mi></mrow></mtd></mtr></mtable></mrow></munder><mspace width="0.2em" /><mrow><mi mathvariant="normal">min</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>,</mo><msub><mi>y</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>[<reflink idref="bib21" id="ref48">21</reflink>] and [<reflink idref="bib14" id="ref49">14</reflink>] provide theoretical justifications for the use of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> to reduce the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> . As can be seen in the third section, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> may result in serious biases for small Gini indices, but this problem can be easily solved by replacing <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> in equation (<reflink idref="bib4" id="ref50">4</reflink>) with the smooth (or midpoint) distribution function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo>*</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><mo stretchy="false">[</mo><mrow><mi>δ</mi><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo><</mo><mi>t</mi></mrow><mo stretchy="false">)</mo><mo>+</mo><mn>0.5</mn><mi>δ</mi><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mo stretchy="false">]</mo></math> </ephtml> , and the resulting estimator coincides with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> (see [<reflink idref="bib4" id="ref51">4</reflink>]). In addition, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> are related when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>i</mi></msub><mo>≠</mo><msub><mi>y</mi><mi>j</mi></msub></math> </ephtml> for all <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>≠</mo><mi>j</mi></math> </ephtml> , since <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>if this condition is satisfied, and <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mi>n</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mrow><mo>(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac></mrow></mstyle></mrow><mo>)</mo></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p>according to equations (<reflink idref="bib7" id="ref52">7</reflink>) and (<reflink idref="bib8" id="ref53">8</reflink>). Note that expressions (<reflink idref="bib5" id="ref54">5</reflink>) and (<reflink idref="bib7" id="ref55">7</reflink>), or their equivalent formulations, are more frequently used in practice, and practitioners must be aware of the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> when <emph>G</emph> is small. The use of the smooth distribution function in equation (<reflink idref="bib4" id="ref56">4</reflink>) will prevent this bias problem. For empirical distributions, additional formulations of the Gini index can be seen in [<reflink idref="bib23" id="ref57">23</reflink>].</p> <p>For a finite population <emph>U</emph> with <emph>N</emph> individuals, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>Y</mi><mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>U</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> denotes a sequence of non-negative random variables with the same distribution function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>U</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> are the population values of the variable of interest. In practice, social surveys are used to estimate the Gini index, and they are generally based on complex sampling designs with unequal probabilities. Therefore, the sample <emph>S</emph> is now selected from <emph>U</emph> by using a sampling design with survey weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mi>i</mi></msub><mo>=</mo><msubsup><mi>π</mi><mi>i</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mi>i</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> are the inclusion probabilities, with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>∈</mo><mi>S</mi></math> </ephtml> . The problem of estimating <emph>G</emph> from finite populations thus entails two steps. First, an empirical version of <emph>G</emph> based on the population values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>U</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> is required. We denote the population empirical versions of <emph>G</emph> in finite populations as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>c</mi></msubsup></math> </ephtml> , and they are defined as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , respectively, after substituting the sample values with the population values in equations (<reflink idref="bib4" id="ref58">4</reflink>), (<reflink idref="bib5" id="ref59">5</reflink>) and (<reflink idref="bib7" id="ref60">7</reflink>). The second step is to estimate the selected population empirical version ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>b</mi></msubsup></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>c</mi></msubsup></math> </ephtml> ) using weighted estimators. Some that can be found in the literature are: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>a</mi></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>2</mn><mrow><mrow><mover><mi>N</mi><mo stretchy="false">^</mo></mover></mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow><mi>w</mi></msub></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>w</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub><msub><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi></msub><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo>;</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>b</mi></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mrow><mrow><mover><mi>N</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mn>2</mn></msup><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow><mi>w</mi></msub></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mspace width=".1em" /><mi>j</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>w</mi><mi>i</mi></msub><msub><mi>w</mi><mi>j</mi></msub><mo fence="false" stretchy="false">|</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><msub><mi>y</mi><mi>j</mi></msub></mrow><mo fence="false" stretchy="false">|</mo><mo>;</mo></mstyle></math> </ephtml> </p> <p>Graph</p> <p>and <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup><mo>=</mo><mn>1</mn><mo>−</mo><mrow><mfrac><mrow><msub><mrow><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi></msub></mrow><mrow><msub><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi></msub></mrow></mfrac></mrow><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>N</mi><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>w</mi><mi>i</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi></msub><mo>=</mo><msup><mrow><mover><mi>N</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>w</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi></msub><mo>=</mo><msup><mrow><mover><mi>N</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>w</mi><mi>i</mi></msub><msub><mi>z</mi><mrow><mi>i</mi><mo>.</mo><mi>w</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mrow><mi>i</mi><mo>.</mo><mi>w</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><mrow><mover><mi>N</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><msub><mi>w</mi><mi>i</mi></msub></mrow></mfrac></mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msup><mrow /><mrow><mspace width=".1em" /><mi>j</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></msup></mtd></mtr><mtr><mtd><msup><mrow /><mrow><mspace width=".1em" /><mi>j</mi><mo>≠</mo><mi>i</mi></mrow></msup></mtd></mtr></mtable></mrow></munder><mspace width="0.2em" /><msub><mi>w</mi><mi>j</mi></msub><mrow><mi mathvariant="normal">min</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>,</mo><msub><mi>y</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mover><mi>N</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>w</mi><mi>i</mi></msub><mi>δ</mi><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>≤</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> . Note that equations (<reflink idref="bib10" id="ref61">10</reflink>), (<reflink idref="bib11" id="ref62">11</reflink>) and (<reflink idref="bib12" id="ref63">12</reflink>) reduce, respectively, to equations (<reflink idref="bib4" id="ref64">4</reflink>), (<reflink idref="bib5" id="ref65">5</reflink>) and (<reflink idref="bib7" id="ref66">7</reflink>) under simple random sampling without replacement (SRSWOR).</p> <hd id="AN0182244234-6">Simulation Studies to Analyse the Bias</hd> <p>In this section, we analyse the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> in comparison to the true (asymptotic) value <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> , and using samples selected from infinite populations. This analysis is equivalent to the problem of analysing the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>c</mi></msubsup></math> </ephtml> in comparison to the true (asymptotic) value <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> .</p> <hd id="AN0182244234-7">Description</hd> <p>We consider various continuous probabilistic distributions (Pareto, Dagum, Lognormal, Weibull and Gamma) often used in the modelling of income distributions. For each probabilistic distribution, parameters involved in the theoretical formulation of <emph>G</emph> are selected such that <emph>G</emph> takes the values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><mn>0.1</mn><mo>,</mo><mn>0.2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mn>0.8</mn></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> , thus allowing us to examine different levels of inequality. Additional parameters required in distributions are also fixed, and all of them can be seen in the Supplemental material (Table A1). For the Dagum distribution, the theoretical value of <emph>G</emph> depends on both shape parameters <emph>a</emph> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> , and for this reason the values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mrow><mn>0.5</mn><mo>,</mo><mn>20</mn></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> are also fixed, and such distributions are denoted, respectively, as Dagum-p0.5 and Dagum-p20. The aim is to analyse the biases of the various estimators of the Gini index under the described scenarios, with these estimators being calculated using samples randomly drawn from an underlying continuous distribution with a true value <emph>G</emph> for the Gini index. This framework is also adopted by [<reflink idref="bib15" id="ref67">15</reflink>], [<reflink idref="bib14" id="ref68">14</reflink>], [<reflink idref="bib5" id="ref69">5</reflink>], etc. We analyse both small and large sample sizes, specifically, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mrow><mn>50</mn><mo>,</mo><mn>500</mn></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> . This study is equivalent to analysing the biases for samples, with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> </ephtml> , selected under SRSWOR from a large finite population ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo stretchy="false">→</mo><mi mathvariant="normal">∞</mi></math> </ephtml> ), with population values drawn from the analysed probabilistic distributions.</p> <p>Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> be a given statistic for the unknown parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> </ephtml> , based on the observations <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>S</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> . Throughout this article, the expected value based on <emph>R</emph> replications of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msubsup><mrow><mover><mi>θ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac></mrow><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></munderover><mspace width="0.2em" /><msubsup><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> is the statistic <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> evaluated at the <emph>r</emph>th pseudo original sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msup></math> </ephtml> , which is also selected, with size <emph>n</emph>, from the distribution function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math> </ephtml> . <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>1000</mn></math> </ephtml> replications are considered in simulation studies. The empirical measures can be expressed in terms of either the true Gini index <emph>G</emph> or the expected values of estimators. We use the expected values because large biases can be obtained and <emph>G</emph> is unknown in practice. Reporting the results in terms of <emph>G</emph> makes it more difficult for empirical researchers to assess the performance of estimators for the specific data that they are analysing. Finally, note that various figures in this article require only a customary estimator of <emph>G</emph>, and we use <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mspace width="0.25em" /></math> </ephtml> because it is less biased than its competitors ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> ).</p> <p>In this section, we first use Monte Carlo simulations to investigate the relative bias ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> ) of the various empirical versions ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> ) in comparison to the true (asymptotic) value <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> . For a given statistic <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> , this measure is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi><mo>=</mo><mn>100</mn><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>B</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo></mrow><mi>θ</mi></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where the empirical bias is given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo><mo>=</mo><msub><mi>E</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo><mo>−</mo><mi>θ</mi><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msubsup><mrow><mover><mi>θ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup><mspace width=".1em" /><mo>−</mo><mspace width=".1em" /><mi>θ</mi></math> </ephtml> . Comparisons are based on distributions with different levels of skewness because the value of the coefficient of skewness may have an impact on the bias of estimators of the Gini index. For discrete distributions, the coefficient of skewness is defined as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>μ</mi><mrow><mi>n</mi><mn>.3</mn></mrow></msub></mrow><mrow><msubsup><mi>σ</mi><mi>n</mi><mn>3</mn></msubsup></mrow></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>n</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mrow><msub><mi>μ</mi><mrow><mi>n</mi><mn>.2</mn></mrow></msub></mrow><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math> </ephtml> is the sample standard deviation, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mrow><mi>n</mi><mo>.</mo><mi>α</mi></mrow></msub><mo>=</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow><msup><mo stretchy="false">)</mo><mi>α</mi></msup></math> </ephtml> the <emph>α</emph>th central moment based on <emph>S</emph>. The aim of Figure 1 is to investigate the skewness for the probabilistic distributions considered in this article, so this figure displays the expected values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup></math> </ephtml> versus the expected values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> are calculated using equation (<reflink idref="bib13" id="ref70">13</reflink>) after substituting <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> , respectively, and which are computed using equations (<reflink idref="bib14" id="ref71">14</reflink>) and (<reflink idref="bib7" id="ref72">7</reflink>) at the <emph>r</emph>th pseudo original sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msup></math> </ephtml> .</p> <p>Graph: Figure 1. Expected values of the coefficient of skewness (γ¯nR) based on samples with sizes n={50,500}, and randomly selected from various continuous probabilistic distributions (infinite populations). The x-axes show the expected values of the estimator G^nc (G¯nc.R).</p> <p>From Figure 1 we observe that the Pareto distribution is the most highly skewed distribution, followed by the Dagum-p20, Dagum-p0.5 and Lognormal distributions, in that order. The Weibull and Gamma distributions have similar values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup></math> </ephtml> , and they are the least skewed distributions in this study. For highly skewed distributions, serious biases can be observed when <emph>G</emph> is large, as a result of which the maximum value of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> is far from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mn>0.8</mn></math> </ephtml> , the maximum Gini index used in this study. This is not the case with less skewed distributions, since accurate estimates are also obtained when <emph>G</emph> is large, and the expected values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> are close to the required Gini index. For the various probabilistic distributions, the expected skewness increases as the sample size rises. This can be explained by the upper bound for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> based on the sample size and suggested by [<reflink idref="bib13" id="ref73">13</reflink>]. This bound indicates that estimates for the coefficient of skewness may underestimate the true value when the sample size is small ([<reflink idref="bib17" id="ref74">17</reflink>]). For the Dagum distribution, the expected skewness increases as its shape parameter <emph>p</emph> increases.</p> <p>In this section we also analyse the impact of the skewness on the bias using box plots for estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> . Thus, we illustrate this relationship between skewness and bias by comparing two different distributions in terms of skewness (Pareto and Gamma, as can be seen in Figure 1).</p> <hd id="AN0182244234-8">Results and Conclusions</hd> <p>Figure 2 displays the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi><mi>s</mi></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>50</mn></math> </ephtml> . First, we analyse the results from the less skewed distributions (Weibull and Gamma). The bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> is negligible for the various expected values of estimators. The bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> is slightly larger, in absolute terms, than that of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , but lies within a reasonable range. Biases of both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> do not seem to be affected by the value of the Gini index. <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> is severely biased when the expected values of estimators are small, with values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> that can be close to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mtext>%</mtext></math> </ephtml> . This empirical version must be modified to correct this bias, and two simple solutions are discussed in "Estimation" subsection. First, the distribution function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> can be replaced, in equation (<reflink idref="bib4" id="ref75">4</reflink>), by the smooth distribution function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo>*</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> . This adjustment allows empirical versions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> to be equivalent. Second, we can use one of the transformations described in equations (<reflink idref="bib8" id="ref76">8</reflink>) and (<reflink idref="bib9" id="ref77">9</reflink>) when all observations are different. As the Gini index increases, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> decreases and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> have similar <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi><mi>s</mi></math> </ephtml> .</p> <p>Graph: Figure 2. Relative biases (RBs) of estimators G^na, G^nb, and G^nc based on samples with size n=50, and randomly selected from various continuous probabilistic distributions (infinite populations).</p> <p>For heavy-tailed distributions (Pareto, Dagum-p20, Dagum-p0.5 and Lognormal), biases of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> seem to be affected by the value of the Gini index, and serious negative <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi><mi>s</mi></math> </ephtml> are obtained as the expected values of estimators increase (as much as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>25</mn><mtext>%</mtext></math> </ephtml> ). We also observe a strong relationship between the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> and the coefficient of skewness. The largest <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi><mi>s</mi></math> </ephtml> , in absolute terms, are produced by the Pareto distribution, which is the most skewed (see Figure 1), and biases, in absolute terms, decrease as the values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup></math> </ephtml> decrease.</p> <p>For larger sample sizes, readers are referred to the Supplemental material, where Figure A1 replicates Figure 2 for samples with size <emph>n</emph> = 500. We point out that the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> , in absolute terms, decreases as the sample size increases. For less skewed distributions, the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> is negligible, and the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> can be close to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mtext>%</mtext></math> </ephtml> for the various distributions. Non-negligible biases are also observed for heavy-tailed distributions, with values of <emph>RB</emph> close to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>15</mn><mtext>%</mtext></math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>500</mn></math> </ephtml> .</p> <p>In Figure 3 we investigate the effect of the skewness on the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> using box plots and various Gini indices, with the most (Pareto) and the least (Gamma) skewed distributions from this study. From Figure 2 we observe that the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> is negligible for the various Gini indices when samples are selected from the Gamma distribution, while Figure 3 confirms that the estimates are concentrated, with a low variability, around the target value <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> . This is not the case with the Pareto distribution, which shows highly biased estimates and marked variability. From Figures 2 and 3 we observe that the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , in absolute terms, increases as the Gini index rises, while from Figure 3 we see that the variability of estimates also becomes higher as <emph>G</emph> increases, with values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> that can be larger than 0.9 when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mn>0.4</mn></math> </ephtml> , or smaller than 0.3 when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mn>0.8</mn></math> </ephtml> .</p> <p>Graph: Figure 3. Box plots for 1000 estimates of the Gini index (G) using the estimator G^nc and various values of G. Samples, with size n=50, are randomly selected from the Pareto and Gamma distributions (infinite populations).</p> <hd id="AN0182244234-9">Bias Correction Procedures</hd> <p>Results from the third section indicate that the customary estimators of the Gini index can be severely biased, especially for heavy-tailed distributions and large Gini indices, and that the use of a bias correction procedure may alleviate this bias problem. Various bias correction procedures are presented in this section, before being analysed in the fifth section, and then applied to various real data sets in the seventh section. The sixth section describes a criterion for deciding when to use bias corrections.</p> <p>Bootstrap and jackknife techniques (see [<reflink idref="bib19" id="ref78">19</reflink>], and [<reflink idref="bib57" id="ref79">57</reflink>]) can be used as bias correction procedures. Some authors who have demonstrated the capacity of these methods to correct biases are [<reflink idref="bib41" id="ref80">41</reflink>] and [<reflink idref="bib28" id="ref81">28</reflink>]. When it comes to the problem of estimating the Gini index, such statistical techniques have been used mainly for the construction of confidence intervals and variance estimation (see [<reflink idref="bib34" id="ref82">34</reflink>], and [<reflink idref="bib31" id="ref83">31</reflink>], for bootstrap techniques, and [<reflink idref="bib4" id="ref84">4</reflink>], and [<reflink idref="bib14" id="ref85">14</reflink>], for jackknife techniques). The estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> emerges as the bias corrected version of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> ([<reflink idref="bib14" id="ref86">14</reflink>]; [<reflink idref="bib15" id="ref87">15</reflink>]), but the third section shows that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> can also be severely biased. [<reflink idref="bib51" id="ref88">51</reflink>] investigated a bias correction method for the Gini index, but focussing on the bias due to grouped data. We now explore correction procedures for the biases discussed in the third section. Bias corrections are applied to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup></math> </ephtml> (defined, respectively, for infinite and finite populations) because they are less biased than the alternative empirical versions described in the second section. However, bias correction procedures can also be applied to any other empirical version.</p> <p>For an infinite population, we first suggest the jackknife technique proposed by [<reflink idref="bib38" id="ref89">38</reflink>], which can be easily implemented by a fast algorithm. [<reflink idref="bib29" id="ref90">29</reflink>] showed that this method has desirable properties for the variance estimation of the Gini index. [<reflink idref="bib38" id="ref91">38</reflink>] proposed the application of the jackknife technique on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup></math> </ephtml> , with jackknife estimates defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>2</mn><mrow><mi>n</mi><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow><mo>−</mo><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mfrac></mrow><mrow><mo>{</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></mfrac></mrow><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mspace width="0.2em" /><mi>i</mi><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mrow><mi>n</mi><mo stretchy="false">(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>n</mi><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow><mo>−</mo><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mspace width="0.2em" /><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><mi>k</mi><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></mstyle></mstyle></mstyle></mrow><mo>}</mo></mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mo stretchy="false">(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>,</mo></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> is the regression coefficient defined by equation (<reflink idref="bib6" id="ref92">6</reflink>). Note that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> is equivalent to applying <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup></math> </ephtml> successively to the observations <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>S</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> and after removing the <emph>k</emph>th unit. The bias corrected estimator applied to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup></math> </ephtml> and based on Ogwang's jackknife is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2.</mn><mi>J</mi><mi>o</mi></mrow></msubsup><mo>=</mo><mi>n</mi><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup><mo>−</mo><mo stretchy="false">(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo stretchy="false">)</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2.</mn><mi>J</mi></mrow></msubsup><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2.</mn><mi>J</mi></mrow></msubsup><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mspace width="0.2em" /><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>b</mi><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>.</mo></math> </ephtml> The bias corrected estimator applied to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> and based on jackknife is given by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup><mo>=</mo><mi>n</mi><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>−</mo><mo stretchy="false">(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo stretchy="false">)</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi></mrow></msubsup><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi></mrow></msubsup><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mspace width="0.2em" /><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> is the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> computed from the observations <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msub><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mi>S</mi></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> after removing the <emph>k</emph>th unit. Note that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup></math> </ephtml> is one of the two bias corrected estimators that we report in the results from infinite populations. [<reflink idref="bib41" id="ref93">41</reflink>] proposed an empirical bootstrap bias correction procedure based on pseudo original and bootstrap samples selected from plausible parameters. This method was used to estimate the prediction mean square error in small area estimation of proportions. We also propose the adaption of this empirical bootstrap method to the problem of estimating the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , thus giving rise to a novel bias corrected estimator of <emph>G</emph>.</p> <p>The empirical bootstrap procedure considers a set of plausible parameters, which are randomly generated from a confidence interval for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> . For each plausible parameter, a pseudo original sample is generated from the underlying distribution of the original sample data. This method uses a cross-validation procedure that splits the various pseudo original samples into two groups: training and validation. In the third section we observed that both the Gini index and the coefficient of skewness have an impact on the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> for heavy-tailed distributions. For the training group, we suggest various functions underlying the bias correction, which depend on the estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> computed from each pseudo original sample and on the expected values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> based on bootstrap samples. [<reflink idref="bib19" id="ref94">19</reflink>] and [<reflink idref="bib25" id="ref95">25</reflink>] indicate that bias corrections may increase the variance, so the validation group is used to choose the optimum bias function that minimises the mean square error (<emph>MSE</emph>) of the suggested bias corrected estimator. In the fifth section, we also investigate the impact of using bias corrections on the <emph>MSE</emph>. For an infinite population, the algorithm for estimating the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> and for computing the suggested bias corrected estimator is described in detail as follows:</p> <p>Step 1 (Plausible parameters). Select at random <emph>H</emph> plausible values for the target parameter <emph>G</emph> from a Uniform distribution, that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>h</mi></msub><mo>∼</mo><mi>U</mi><mi>n</mi><mo stretchy="false">(</mo><mrow><msub><mi>G</mi><mi>L</mi></msub><mo>,</mo><msub><mi>G</mi><mi>U</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> , with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>H</mi></math> </ephtml> , and where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>L</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>U</mi></msub></math> </ephtml> are, respectively, the lower and upper limits of a confidence interval for the true Gini index <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> .</p> <p>Step 2 (Pseudo original samples for training and validation groups). Generate a pseudo original sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>h</mi></msub></math> </ephtml> , with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> </ephtml> , from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>;</mo><msub><mi>G</mi><mi>h</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> and for each <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>H</mi></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>;</mo><msub><mi>G</mi><mi>h</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> is the probability density function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math> </ephtml> with a Gini index equal to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>h</mi></msub></math> </ephtml> . Then, split the <emph>H</emph> samples at random into two groups, the training group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>T</mi></msub></math> </ephtml> and the validation group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>V</mi></msub></math> </ephtml> , such that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>T</mi></msub></math> </ephtml> contains a set of <emph>T</emph> samples ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>t</mi></msub></math> </ephtml> , with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>T</mi></math> </ephtml> ), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>V</mi></msub></math> </ephtml> contains <emph>V</emph> samples ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>v</mi></msub></math> </ephtml> , with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>V</mi></math> </ephtml> ) and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mi>T</mi><mo>+</mo><mi>V</mi></math> </ephtml> .</p> <p>Step 3 (Estimates from the pseudo original samples). For the training group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>T</mi></msub></math> </ephtml> , compute <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow></msub></math> </ephtml> for each sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>t</mi></msub></math> </ephtml> , and using, respectively, equations (<reflink idref="bib7" id="ref96">7</reflink>) and (<reflink idref="bib14" id="ref97">14</reflink>). Similarly, for the validation group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>V</mi></msub></math> </ephtml> , compute <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow></msub></math> </ephtml> using the samples <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>v</mi></msub></math> </ephtml> .</p> <p>Step 4 (Training phase: expected values based on bootstrap samples). For the training group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>T</mi></msub></math> </ephtml> , generate <emph>B</emph> bootstrap samples <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>S</mi><mi>t</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> , with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> </ephtml> , from each sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>t</mi></msub></math> </ephtml> , with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>B</mi></math> </ephtml> . Compute <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> for each bootstrap sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>S</mi><mi>t</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> , using equations (<reflink idref="bib7" id="ref98">7</reflink>) and (<reflink idref="bib14" id="ref99">14</reflink>), respectively. The expected values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> based on bootstrap samples are denoted, respectively, as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mi>B</mi></msubsup></math> </ephtml> , and are computed using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace width="0.25em" /></math> </ephtml> in equation (<reflink idref="bib13" id="ref100">13</reflink>), after substituting <emph>r</emph> and <emph>R</emph> with <emph>b</emph> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> , respectively.</p> <p>Step 5 (Training phase: expected values based on pseudo original samples). For the training group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>T</mi></msub></math> </ephtml> , generate <emph>R</emph> pseudo original samples <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>S</mi><mi>t</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> , with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> </ephtml> , from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>;</mo><msub><mi>G</mi><mi>t</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> and for each <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>T</mi></math> </ephtml> , with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mrow><mi mathvariant="normal">R</mi></mrow></math> </ephtml> . Compute <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> for each sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>S</mi><mi>t</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> , and using equation (<reflink idref="bib7" id="ref101">7</reflink>). The expected value based on pseudo original samples is denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> , and is computed using equation (<reflink idref="bib13" id="ref102">13</reflink>).</p> <p>Step 6 (Training phase: coefficient estimates). For the training group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>T</mi></msub></math> </ephtml> , estimate the unknown coefficients of a set of eligible bias functions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mi>c</mi></msubsup><mo>;</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup><mo>;</mo><msub><mrow><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mi>n</mi><mi>t</mi></mrow></msub><mo>;</mo><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mi>B</mi></msubsup></mrow><mo stretchy="false">)</mo></math> </ephtml> , with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>L</mi></math> </ephtml> , that predict the variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mi>t</mi></msub><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>−</mo><msub><mi>G</mi><mi>t</mi></msub></math> </ephtml> . An example of bias function is the linear expression <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>−</mo><msub><mi>G</mi><mi>t</mi></msub><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mi>B</mi></msubsup><mo>−</mo><msub><mrow><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mi>n</mi><mi>t</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>Step 7 (Validation phase: bias corrected estimators). For the validation group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>V</mi></msub></math> </ephtml> and for each function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>l</mi></msub></math> </ephtml> , compute the suggested bias corrected estimator of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>v</mi></msub></math> </ephtml> , defined by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow><mi>c</mi></msubsup><mo>−</mo><msub><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover></mrow><mi>l</mi></msub><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow><mi>c</mi></msubsup><mo>;</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup><mo>;</mo><msub><mrow><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mi>n</mi><mi>v</mi></mrow></msub><mo>;</mo><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow><mi>B</mi></msubsup></mrow><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover></mrow><mi>l</mi></msub></math> </ephtml> is the bias function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>l</mi></msub></math> </ephtml> after substituting its coefficients with the estimates computed in Step 6.</p> <p>Step 8 (Validation phase: optimum function). For the validation group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ω</mi><mi>V</mi></msub></math> </ephtml> , identify the optimum function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>o</mi><mi>p</mi><mi>t</mi></mrow></msub></math> </ephtml> that minimises the <emph>MSE</emph> of the estimators <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></math> </ephtml> , and which is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>S</mi><msub><mi>E</mi><mi>l</mi></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mi>V</mi></mfrac></mrow><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><mspace width="0.2em" /><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>v</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>G</mi><mi>v</mi></msub></mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>Step 9 (Bias corrected estimator). Compute the suggested bias corrected estimator <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>−</mo><msub><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>o</mi><mi>p</mi><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>;</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup><mo>;</mo><msub><mrow><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub><mo>;</mo><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>B</mi></msubsup></mrow><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>B</mi></msubsup></math> </ephtml> are, respectively, the expected values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> based on bootstrap samples and derived from the original sample <emph>S</emph>.</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> is the second bias corrected estimator computed for infinite populations. Any method can be used to construct the confidence interval required in Step 1. Some existing confidence intervals for the Gini index are based on bootstrap (Qin, Rao, and Wu 2010), jackknife ([<reflink idref="bib4" id="ref103">4</reflink>]), linearisation ([<reflink idref="bib16" id="ref104">16</reflink>]) or empirical likelihood ([<reflink idref="bib5" id="ref105">5</reflink>]). Variance estimators for the Gini index ([<reflink idref="bib29" id="ref106">29</reflink>]) can also be used to construct confidence intervals based on the normality assumption. For the sake of simplicity, we use the traditional bootstrap method with confidence interval limits given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>L</mi></msub><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mo stretchy="false">(</mo><mrow><mn>0.005</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>U</mi></msub><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mo stretchy="false">(</mo><mrow><mn>0.995</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> is the <emph>α</emph>th quantile of the bootstrap estimates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> . The latter are computed using the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> on the bootstrap sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></math> </ephtml> , which is taken, with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> </ephtml> , from the original sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> . As discussed by [<reflink idref="bib41" id="ref107">41</reflink>], the selected interval must be broad enough to contain the target parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> . We use a confidence level of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>99</mn><mtext>%</mtext></math> </ephtml> because of the serious biases detected in the third section. [<reflink idref="bib41" id="ref108">41</reflink>] also argue that the size of this confidence interval has no direct effect on the bound of the bias, and give a discussion on the number of parameters that should be included in the training and validation groups.</p> <p>For the training group, Step 6 requires bias functions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>l</mi></msub></math> </ephtml> that predict <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mi>t</mi></msub><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>−</mo><msub><mi>G</mi><mi>t</mi></msub></math> </ephtml> with the aim of estimating the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo stretchy="false">[</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></mrow><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></mrow><mo stretchy="false">]</mo><mo>−</mo><mi>G</mi></math> </ephtml> . In the third section, we concluded that both the Gini index and the coefficient of skewness may have an impact on the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , so we suggest bias functions that depend on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow></msub></math> </ephtml> . The expected values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>n</mi><mi>t</mi></mrow><mi>B</mi></msubsup></math> </ephtml> based on bootstrap samples are also considered. Table A2 from the Supplemental material describes the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mn>7</mn></math> </ephtml> candidate bias functions considered in Step 6 of the suggested algorithm. For the sake of simplicity, we only consider multiple linear regression functions, but more complex functions and/or additional statistics can also be used, and are expected to yield more accurate results.</p> <p>For an infinite population, additional bias correction procedures can be computed (see [<reflink idref="bib57" id="ref109">57</reflink>]). For instance, we also calculated, in the fifth section, bootstrap methods based on additive and multiplicative corrections (see [<reflink idref="bib25" id="ref110">25</reflink>], and [<reflink idref="bib41" id="ref111">41</reflink>], for detailed definitions), but we omitted them because the bias correction estimators (<reflink idref="bib15" id="ref112">15</reflink>) and (<reflink idref="bib17" id="ref113">17</reflink>) are less biased. [<reflink idref="bib41" id="ref114">41</reflink>] also argue that the aforementioned additive and multiplicative corrections may yield non-negligible biases with small samples, meaning alternative bias correction procedures may be preferable.</p> <p>Bootstrap and jackknife techniques were originally designed for infinite populations, and do not have a direct application to finite populations due to the inherent features of survey sampling. Adjustments are thus required to apply these methods to finite populations ([<reflink idref="bib44" id="ref115">44</reflink>]:71). For finite populations, the rescaled bootstrap technique ([<reflink idref="bib45" id="ref116">45</reflink>]) can be used for bias correction of a given empirical version of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> . This method has been used in many research studies ([<reflink idref="bib6" id="ref117">6</reflink>]; [<reflink idref="bib35" id="ref118">35</reflink>]; etc.) in many areas (see [<reflink idref="bib36" id="ref119">36</reflink>]; [<reflink idref="bib59" id="ref120">59</reflink>]; etc.). Simplicity is the main advantage of the rescaled bootstrap over alternative bootstrap methods, which can be more computationally intensive. The rescaled bootstrap consists in computing a new set of weights (named bootstrap weights) for each bootstrap sample, which are obtained by applying a scale adjustment to the original survey weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mi>i</mi></msub></math> </ephtml> . Specifically, the bootstrap weights are given by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msub><mi>w</mi><mi>i</mi></msub><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>r</mi><mi>i</mi></msub><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>∈</mo><mi>S</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>B</mi></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>i</mi></msub></math> </ephtml> denotes the number of times that <emph>i</emph>th unit is selected in the bootstrap sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></math> </ephtml> . For a finite population, we first consider the additive bias corrected estimator of <emph>G</emph> based on the rescaled bootstrap, which is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>a</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup><mo>−</mo><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup><mo>−</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> is the expected value of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup></math> </ephtml> based on the bootstrap estimates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> , and which are defined as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup></math> </ephtml> after substituting the original survey weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mi>i</mi></msub></math> </ephtml> with the bootstrap weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> .</p> <p>Second, we also consider the aforementioned empirical bootstrap bias correction. We now describe an extension of this method to finite populations. It requires the rescaled bootstrap technique along with confidence intervals and estimators based on survey weights. A confidence interval that can be used in Step 1 is given by the limits <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>L</mi></msub><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mo stretchy="false">(</mo><mrow><mn>0.005</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>U</mi></msub><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mo stretchy="false">(</mo><mrow><mn>0.995</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> is the <emph>α</emph>th quantile of the weighted estimates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> derived from the rescaled bootstrap and based on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup></math> </ephtml> . The following Step 1-b must be included between Steps 1 and 2:</p> <p>Step 1-b (Pseudo original finite population). Generate a pseudo original population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>U</mi><mi>k</mi><mo>*</mo></msubsup></math> </ephtml> with observations given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mrow><msubsup><mi>y</mi><mi>i</mi><mo>*</mo></msubsup><mo>:</mo><mi>i</mi><mo>∈</mo><msubsup><mi>U</mi><mi>k</mi><mo>*</mo></msubsup></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> and selected from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>;</mo><msub><mi>G</mi><mi>k</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> , with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>K</mi></math> </ephtml> .</p> <p>The pseudo original samples of Steps 2 and 5 are selected from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>U</mi><mi>k</mi><mo>*</mo></msubsup></math> </ephtml> instead of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>;</mo><msub><mi>G</mi><mi>k</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> , and using the same sampling design as for the original sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> . The weighted coefficient of skewness is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>μ</mi><mrow><mi>w</mi><mn>.3</mn></mrow></msub></mrow><mrow><msubsup><mi>σ</mi><mi>w</mi><mn>3</mn></msubsup></mrow></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>w</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mrow><msub><mi>μ</mi><mrow><mi>w</mi><mn>.2</mn></mrow></msub></mrow><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mrow><mi>w</mi><mo>.</mo><mi>α</mi></mrow></msub><mo>=</mo><msup><mrow><mover><mi>N</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mspace width="0.2em" /><msub><mi>w</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">¯</mo></mover></mrow></mrow><mi>w</mi></msub></mrow><msup><mo stretchy="false">)</mo><mi>α</mi></msup></math> </ephtml> . In Step 3, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>h</mi></mrow><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>n</mi><mi>h</mi></mrow></msub></math> </ephtml> are replaced by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>h</mi></mrow><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>h</mi></mrow></msub></math> </ephtml> , which are calculated using the sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>h</mi></msub></math> </ephtml> and equations (<reflink idref="bib12" id="ref121">12</reflink>) and (<reflink idref="bib19" id="ref122">19</reflink>), respectively. In Step 4, bootstrap estimates are substituted with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> , which are obtained using the rescaled bootstrap method. The expected values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi></msub></math> </ephtml> based on the rescaled bootstrap are denoted, respectively, as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mi>B</mi></msubsup></math> </ephtml> , and they are computed using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace width="0.25em" /></math> </ephtml> in equation (<reflink idref="bib13" id="ref123">13</reflink>), after substituting <emph>r</emph> and <emph>R</emph> with <emph>b</emph> and <emph>B</emph>, respectively. The same set of eligible functions are used in Step 6, but they depend on weighted quantities, that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mi>c</mi></msubsup><mo>;</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup><mo>;</mo><msub><mrow><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mi>w</mi><mi>t</mi></mrow></msub><mo>;</mo><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>w</mi><mi>t</mi></mrow><mi>B</mi></msubsup></mrow><mo stretchy="false">)</mo></math> </ephtml> . In Step 7, the suggested bias corrected estimators of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>v</mi></msub></math> </ephtml> are defined by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>v</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>v</mi></mrow><mi>c</mi></msubsup><mo>−</mo><msub><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover></mrow><mi>l</mi></msub><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>w</mi><mi>v</mi></mrow><mi>c</mi></msubsup><mo>;</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>w</mi><mi>v</mi></mrow><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup><mo>;</mo><msub><mrow><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mi>w</mi><mi>v</mi></mrow></msub><mo>;</mo><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mrow><mi>w</mi><mi>v</mi></mrow><mi>B</mi></msubsup></mrow><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>and they are used to identify, in Step 8, the optimum function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>o</mi><mi>p</mi><mi>t</mi></mrow></msub></math> </ephtml> . Finally, the suggested bias corrected estimator that we compute in finite populations is given by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup><mo>−</mo><msub><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>o</mi><mi>p</mi><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup><mo>;</mo><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi></mrow></msubsup><mo>;</mo><msub><mrow><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>w</mi></msub><mo>;</mo><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>w</mi><mi>B</mi></msubsup></mrow><mo stretchy="false">)</mo><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>For finite populations, additional bias correction procedures can also be computed. For instance, we also calculated, in the fifth section, [<reflink idref="bib10" id="ref124">10</reflink>] jackknife (see [<reflink idref="bib7" id="ref125">7</reflink>]; [<reflink idref="bib4" id="ref126">4</reflink>]) and the multiplicative bootstrap method (see [<reflink idref="bib25" id="ref127">25</reflink>]), but we omitted them because Campbell's jackknife is more biased than equations (<reflink idref="bib18" id="ref128">18</reflink>) and (<reflink idref="bib20" id="ref129">20</reflink>), and additive and multiplicative bootstrap methods give similar results.</p> <hd id="AN0182244234-10">Simulation Studies to Analyse the Bias Correction Procedures</hd> <p></p> <hd id="AN0182244234-11">Description</hd> <p>We now evaluate various bias correction procedures by means of the <emph>RB</emph> measure defined in the third section, and using the probabilistic distributions and sample sizes also described in that section. As discussed in the fourth section, bias correction procedures substantially mitigate the detected biases, but the price to pay is a possible increase in the <emph>MSE</emph>. For this reason, we use the relative root mean square error ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>R</mi><mi>M</mi><mi>S</mi><mi>E</mi></math> </ephtml> ) to investigate the effect of bias corrections on efficiency. For a given statistic <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> , the corresponding <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>R</mi><mi>M</mi><mi>S</mi><mi>E</mi></math> </ephtml> based on <emph>R</emph> replications is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>R</mi><mi>M</mi><mi>S</mi><mi>E</mi><mo>=</mo><mn>100</mn><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>M</mi><mi>S</mi><msub><mi>E</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mi>θ</mi></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where the empirical mean square error is given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>S</mi><msub><mi>E</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo><mo>=</mo><msup><mi>R</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></munderover><mspace width="0.2em" /><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><mi>θ</mi></mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>.</mo></math> </ephtml> The sixth and the eighth sections give more detailed discussions on the importance of both bias and MSE measures. In the third section, we observed that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> yields extremely large values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> when <emph>G</emph> is small, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> is slightly more biased than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> . For the sake of clarity, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> and the corresponding bias corrected estimators are omitted from the figures in this section, but they can also be computed, as discussed in the fourth section. Similarly, for finite populations, weighted estimators of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>a</mi></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>b</mi></msubsup></math> </ephtml> are omitted. For infinite populations, the percentage of the number of times that each eligible bias function is selected as the optimum function of the suggested algorithm described in the fourth section can be seen in the Supplemental material (Table A3). For the various probabilistic distributions, the bias function defined in equation (<reflink idref="bib16" id="ref130">16</reflink>) is the most often selected as the optimum function.</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mn>1000</mn></math> </ephtml> bootstrap samples are used in bootstrap methods. Following [<reflink idref="bib41" id="ref131">41</reflink>], we consider <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>200</mn></math> </ephtml> plausible parameters, of which <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>60</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>140</mn></math> </ephtml> are used for the training and validation groups, respectively. Samples are selected from finite populations with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>10000</mn></math> </ephtml> , which in turn are drawn from the investigated continuous distributions. We consider unequal inclusion probabilities by using the randomised systematic sampling design ([<reflink idref="bib58" id="ref132">58</reflink>]). The effect of the design is increased by generating inclusion probabilities <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mi>i</mi></msub></math> </ephtml> with a correlation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.7</mn></math> </ephtml> between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mi>i</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>i</mi></msub></math> </ephtml> ([<reflink idref="bib5" id="ref133">5</reflink>]).</p> <hd id="AN0182244234-12">Results and Conclusions</hd> <p>For infinite populations, Figure 4 displays the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi><mi>s</mi></math> </ephtml> of estimators of <emph>G</emph> based on samples with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>50</mn></math> </ephtml> . We observe that the bias correction procedures reduce, in absolute terms, the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , although the empirical bootstrap ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> ) performs better than the bias corrected estimator based on the jackknife ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup></math> </ephtml> ), especially for large Gini indices. For the Pareto distribution, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi><mi>s</mi></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> are close to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mtext>%</mtext></math> </ephtml> when their expected values are smaller than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.3</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.4</mn></math> </ephtml> , respectively. For the Dagum-p20 distribution, this situation comes when the expected values are smaller than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.4</mn></math> </ephtml> , and the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi><mi>s</mi></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> are close to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mtext>%</mtext></math> </ephtml> when the expected values are smaller than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.5</mn></math> </ephtml> in the Dagum-p0.5 distribution. For the Lognormal distribution, the bias corrected estimators give negligible biases (absolute values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> smaller than 2%) for the various Gini indices. In the extreme scenario of heavy-tailed distributions and large Gini indices, the values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> may be as much as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>25</mn><mtext>%</mtext></math> </ephtml> (see the Pareto distribution at the last expected value of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> ), and the empirical bootstrap may reduce this <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> by more than 10%. The gain in accuracy of the empirical bootstrap over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup></math> </ephtml> is greater when n = 500 (see Figure A2 in the Supplemental material). For instance, with expected values of estimators close to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.7</mn></math> </ephtml> in the Pareto distribution, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> s of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> are <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>15</mn><mtext>%</mtext></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>12</mn><mtext>%</mtext></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>7</mn><mtext>%</mtext></math> </ephtml> , respectively.</p> <p>Graph: Figure 4. Relative biases (RBs) of estimators G^nc, G^nc.Jo, and G^nc.Bp based on samples with size n=50, and randomly selected from various continuous probabilistic distributions (infinite populations).</p> <p>Values of <emph>RB</emph> based on samples with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>50</mn></math> </ephtml> and derived from finite populations can be seen in Figure 5. We observe that the empirical bootstrap ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> ) performs better than the rescaled bootstrap ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>a</mi></mrow></msubsup></math> </ephtml> ), which in turn minimises the bias problem of the customary estimator ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>w</mi><mi>c</mi></msubsup></math> </ephtml> ). Biases for samples with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>500</mn></math> </ephtml> and derived from finite populations can be seen in the Supplemental material (Figure A3).</p> <p>Graph: Figure 5. Relative biases (RBs) of estimators G^wc, G^wc.Ba, and G^wc.Bp based on samples with size n=50, and randomly selected from a finite population with size N=10000. Population values are randomly drawn from various continuous probabilistic distributions.</p> <p>Figure 6 displays the impact on the efficiency of using corrected estimators. The efficiency of bias correction procedures lies in a reasonable range relative to the efficiency of the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , especially for the correction based on the empirical bootstrap. In particular, the empirical bootstrap performs better than the jackknife. In addition, the empirical bootstrap is slightly less efficient than the customary estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , and is more efficient for large Gini indices. The larger the sample size, the lower the impact on the efficiency of both bias corrected estimators, as can be seen in the Supplemental material Figure A4. Results derived from finite populations, and based on small and large samples, can be seen in the Supplemental material Figures A5 and A6.</p> <p>Graph: Figure 6. Relative root mean square errors (RRMSEs) of estimators G^nc, G^nc.Jo, and G^nc.Bp based on samples with size n=50, and randomly selected from various continuous probabilistic distributions (infinite populations).</p> <hd id="AN0182244234-13">A Simulation-Based Criterion for Deciding When to Use Bias Correction</hd> <p>Results from the third section indicate that the three common empirical versions of <emph>G</emph> can be biased for heavy-tailed distributions, which may be a serious issue when the Gini index is large. An important problem that arises in practice is determining when to use bias correction procedures. Note that the bias and the <emph>MSE</emph> (or equivalently the <emph>RB</emph> and the <emph>RRMSE</emph>) are two relevant measures to evaluate the quality of estimators. However, as noted by [<reflink idref="bib47" id="ref134">47</reflink>]:164), the bias must also be small relative to the standard error, since failure to meet this requirement may result in invalid confidence intervals and/or undesirable coverage probabilities. This ratio between the bias and the standard error is popularly referred to as the bias ratio. For a given statistic <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math> </ephtml> and <emph>R</emph> replications, the empirical bias ratio (<emph>BR</emph>) is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>R</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>B</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi>V</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>where the empirical variance is given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>R</mi></msub><mo stretchy="false">[</mo><mrow><msub><mrow><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mi>n</mi></msub></mrow><mo stretchy="false">]</mo><mo>=</mo><msup><mi>R</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></munderover><mspace width="0.2em" /><mo stretchy="false">(</mo><mrow><msubsup><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mrow><mover><mi>θ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup></mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math> </ephtml> . Like the <emph>MSE</emph>, the <emph>BR</emph> also involves both bias and variance of the estimator. [<reflink idref="bib47" id="ref135">47</reflink>]:41) advise empirical researchers to avoid estimators that are considerably biased, and instead seek out estimators with small biases, and then choose one with a small variance. Following this idea, the suggested criterion consists of analysing both <emph>RB</emph> and <emph>BR</emph> measures for the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , and bias corrections are suggested when non-negligible biases are observed. <emph>RRMSE</emph> values can be used to choose the most efficient bias correction estimator. [<reflink idref="bib47" id="ref136">47</reflink>]:165) indicate that the effect of the bias ratio on the coverage probability can be ignored when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>B</mi><mi>R</mi></mrow><mo fence="false" stretchy="false">|</mo><mo><</mo><mn>0.1</mn></math> </ephtml> , and the use of bias corrected estimators is not justified here. The effect on the coverage probability is not extremely pronounced when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>B</mi><mi>R</mi></mrow><mo fence="false" stretchy="false">|</mo><mo>≤</mo><mn>0.5</mn></math> </ephtml> , but it can be a serious problem otherwise. On the other hand, absolute values of <emph>RB</emph> lower than 2% can be considered negligible, and bias correction procedures are not recommended if this is the case. In summary, bias corrections are suggested when the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> satisfies <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>B</mi><mi>R</mi></mrow><mo fence="false" stretchy="false">|</mo><mo>≥</mo><mn>0.1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>R</mi><mi>B</mi></mrow><mo fence="false" stretchy="false">|</mo><mo>≥</mo><mn>2</mn><mtext>%</mtext></math> </ephtml> .</p> <p>The aim of Figure 7 is to show that the customary estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> can yield poor bias ratios, with bias corrections justified because they substantially minimise this problem. For infinite populations and samples with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>50</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> yields absolute values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>R</mi></math> </ephtml> close to 1.4, and poor coverage probabilities are expected. Furthermore, the vertical lines in Figure 7 indicate the first expected value ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> ) with non-negligible biases, that is, with absolute values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> larger than 2%. We see that the condition imposed by the bias ratio ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>B</mi><mi>R</mi></mrow><mo fence="false" stretchy="false">|</mo><mo><</mo><mn>0.1</mn></math> </ephtml> ) is more demanding than the condition based on the relative bias ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>R</mi><mi>B</mi></mrow><mo fence="false" stretchy="false">|</mo><mo><</mo><mn>2</mn><mtext>%</mtext></math> </ephtml> ), that is, the first value of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> with a <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>B</mi><mi>R</mi></mrow><mo fence="false" stretchy="false">|</mo><mo>≥</mo><mn>0.1</mn></math> </ephtml> is smaller than the first value of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> with a <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>R</mi><mi>B</mi></mrow><mo fence="false" stretchy="false">|</mo><mo>≥</mo><mn>2</mn><mtext>%</mtext><mo>.</mo></math> </ephtml> For example, non-negligible biases are observed for the Pareto distribution when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≅</mo><mn>0.2</mn></math> </ephtml> , and the absolute value of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>R</mi></math> </ephtml> is larger than 0.1 in this situation. These results reveal the presence of a mild bias problem, which can be solved using bias correction procedures, as can be seen in Figures 4 and 7. From Figure 7 we also observe that the <emph>BR</emph> values of the corrected estimators, in absolute terms, are generally smaller than 0.5, and are substantially smaller than those of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> . The desirable properties in terms of both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>R</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>B</mi></math> </ephtml> measures and the negligible impact on the efficiency (see the Pareto distribution in Figure 6 when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>=</mo><mn>2</mn></math> </ephtml> ) indicate that correction procedures are recommended to mitigate the detected biases.</p> <p>Graph: Figure 7. Bias ratios (BRs), in absolute terms, of estimators G^nc, G^nc.Jo, and G^nc.Bp based on samples with size n=50, and randomly selected from various continuous probabilistic distributions (infinite populations). Using the estimator G^nc, horizontal and vertical dotted lines are fixed, respectively, at |BR|=0.1 and at the first expected value with |RB|>2%.</p> <p>In Figure 8 we suggest a simulation-based criterion for deciding when to use bias correction procedures. This method is based on the expected values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup></math> </ephtml> , since the Gini index and the skewness have a direct effect on the bias. Samples, with sizes between 50 and 1000, are drawn from the most skewed probabilistic distributions described in the third section. Using the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , this criterion is based on two conditions: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>B</mi><mi>R</mi></mrow><mo fence="false" stretchy="false">|</mo><mo>≥</mo><mn>0.1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>R</mi><mi>B</mi></mrow><mo fence="false" stretchy="false">|</mo><mo>≥</mo><mn>2</mn><mtext>%</mtext><mo>.</mo></math> </ephtml> A grading scale classifies the non-negligible biases into three categories: mild ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>≤</mo><mo fence="false" stretchy="false">|</mo><mrow><mi>R</mi><mi>B</mi></mrow><mo fence="false" stretchy="false">|</mo><mo><</mo><mn>5</mn></math> </ephtml> ), moderate ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>≤</mo><mo fence="false" stretchy="false">|</mo><mrow><mi>R</mi><mi>B</mi></mrow><mo fence="false" stretchy="false">|</mo><mo><</mo><mn>10</mn></math> </ephtml> ) and severe ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mrow><mi>R</mi><mi>B</mi></mrow><mo fence="false" stretchy="false">|</mo><mo>≥</mo><mn>10</mn></math> </ephtml> ). This scale can be used to identify the scenarios where bias corrections are either weakly or strongly recommended. Thus, while bias is not a serious issue for mild biases, the use of bias correction procedures is suggested to reduce this bias. Bias corrections are highly recommended in the case of moderate biases. The bias is a serious problem in the presence of severe biases, meaning bias corrections are strongly advised.</p> <p>Graph: Figure 8. Grading scale based on the relative biases (RBs) and bias ratios (BRs) of G^nc when samples, with sizes n={50,200,500,1000}, are randomly selected from the Pareto, Dagum-p20, Dagum-p05, and Lognormal distributions (infinite populations). Non-negligible biases with |BR|≥0.1 are considered as mild (2≤|RB|<5), moderate (5≤|RB|<10), and severe (|RB|≥10). The x- and y-axes show the expected values of G^nc and γ^n (G¯nc.R and γ¯nR, respectively).</p> <p>For small sample sizes ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>50</mn><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup><mo>≥</mo><mn>2.6</mn></math> </ephtml> , mild biases are expected when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≥</mo><mn>0.2</mn></math> </ephtml> , approximately. Moderate biases are observed when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>γ</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mi>R</mi></msubsup><mo>≥</mo><mn>3.5</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≥</mo><mn>0.37</mn></math> </ephtml> , and bias corrections are highly recommended in this situation. For samples with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>200</mn></math> </ephtml> , mild biases are expected when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≥</mo><mn>0.39</mn><mo>,</mo></math> </ephtml> and moderate biases are obtained when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≥</mo><mn>0.47</mn></math> </ephtml> . For samples with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>500</mn></math> </ephtml> , mild and moderate biases are expected when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≥</mo><mn>0.48</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≥</mo><mn>0.56</mn></math> </ephtml> , respectively. Finally, for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1000</mn></math> </ephtml> , bias corrections could be applied when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≥</mo><mn>0.49</mn></math> </ephtml> , and moderate biases are observed when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">¯</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>R</mi></mrow></msubsup><mo>≥</mo><mn>0.57</mn></math> </ephtml> . Severe biases are not expected when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>1000</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>≤</mo><mn>0.8</mn></math> </ephtml> (the maximum true Gini index considered in this study). In summary, for small samples sizes (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>50</mn></math> </ephtml> ), bias correction procedures may be required when the estimates of the Gini index are > <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.2</mn></math> </ephtml> , and can be highly recommended when they exceed <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.37</mn></math> </ephtml> . For larger sample sizes (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>500</mn></math> </ephtml> ), mild biases can be expected when the estimates of the Gini index are > <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.48</mn></math> </ephtml> , and bias corrections are highly advisable when the estimates of the Gini index are larger than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.56</mn></math> </ephtml> . For different sample sizes and Gini indices associated with specific data that empirical researchers are analysing, the aim of Figure 8 is to depict the values of the coefficient of skewness that would require the use of bias corrections.</p> <hd id="AN0182244234-14">Applications to Real Data Sets</hd> <p>In this section, bias correction procedures are applied for estimating the Gini index in a total of six subpopulations with sizes between 26 and 503, and derived from three real data sets (see Table 1). A common goal in most surveys is not only to provide estimates for the whole population, but also for specific subpopulations (also named domains). For instance, estimates of unemployment in labour-force surveys are provided at national level, but this information is also of special interest at provincial and local levels. In household surveys, subpopulations are usually created on the basis of household sizes or consumption units. Age, sex and occupational groups are also often used to create subpopulations in many studies.</p> <p>Table 1. Estimates of the Gini Index for Various Subpopulations Derived From the ES-SILC, WBES, and WATER Populations. The Null Hypothesis That Data Come From a Specific Continuous Probabilistic Distribution is Tested Using the KS Goodness of Fit Test, and the Corresponding p -Values are Provided. The Bias Corrected Estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> is Based on Probabilistic Distributions That Fit the Data.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="right" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="left" /><col align="center" /></colgroup><tbody><tr><td>Population</td><td align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">n</mi></math></p></td><td align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub></math></p></td><td align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math></p></td><td align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup></math></p></td><td align="center"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math></p></td><td align="center">Distribution</td><td align="center"><italic>KS p</italic>-value</td></tr><tr><td>ES-SILC</td><td>26</td><td>2.81</td><td>0.518</td><td>0.534</td><td>0.540</td><td>Fisk</td><td>0.85</td></tr><tr><td /><td /><td /><td /><td /><td>0.526</td><td>Lognormal</td><td>0.65</td></tr><tr><td /><td>51</td><td>3.51</td><td>0.476</td><td>0.486</td><td>0.488</td><td>Fisk</td><td>0.99</td></tr><tr><td /><td /><td /><td /><td /><td>0.480</td><td>Lognormal</td><td>0.62</td></tr><tr><td /><td /><td /><td /><td /><td>0.499</td><td>Dagum</td><td>0.11</td></tr><tr><td>WBES</td><td>61</td><td>3.22</td><td>0.505</td><td>0.513</td><td>0.509</td><td>Lognormal</td><td>0.38</td></tr><tr><td /><td /><td /><td /><td /><td>0.516</td><td>Fisk</td><td>0.37</td></tr><tr><td /><td>503</td><td>15.85</td><td>0.734</td><td>0.784</td><td>0.752</td><td>Fisk</td><td>0.57</td></tr><tr><td>WATER</td><td>38</td><td>3.92</td><td>0.358</td><td>0.368</td><td>0.370</td><td>Dagum</td><td>0.07</td></tr><tr><td /><td /><td /><td /><td /><td>0.359</td><td>Weibull</td><td>0.06</td></tr><tr><td /><td /><td /><td /><td /><td>0.366</td><td>Fisk</td><td>0.06</td></tr><tr><td /><td /><td /><td /><td /><td>0.363</td><td>Lognormal</td><td>0.05</td></tr><tr><td /><td>74</td><td>3.09</td><td>0.435</td><td>0.439</td><td>0.442</td><td>Fisk</td><td>0.10</td></tr><tr><td /><td /><td /><td /><td /><td>0.442</td><td>Dagum</td><td>0.10</td></tr></tbody></table> </ephtml> </p> <p>1 KS= Kolmogorov-Smirnov; ES-SILC= extracted from the 2019 Spanish Survey on Income and Living Conditions; WBES= World Bank's Enterprise Survey.</p> <p>The first real data set consists of total net household incomes extracted from the 2019 Spanish Survey on Income and Living Conditions (ES-SILC). Subpopulations, with sizes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mrow><mn>26</mn><mo>,</mo><mn>51</mn></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> , are created using different consumption units, with the aim of using the Gini index to estimate income inequality. The second real data set is obtained from the World Bank's Enterprise Survey (WBES), which has been used extensively in international management studies ([<reflink idref="bib24" id="ref137">24</reflink>]; [<reflink idref="bib52" id="ref138">52</reflink>]; etc.). For private sector firms from over 130 developed and developing countries, the WBES contains information on a broad range of topics including competition performance, corruption, financial data, infrastructure, technology, etc. Using this survey, we estimate the Gini index of the labour productivity per hour worked in Argentinean firms for the years 2017 and 2018. The sizes of the resulting subpopulations are <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mrow><mn>61</mn><mo>,</mo><mn>503</mn></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> . Finally, the third real data set (named WATER) consists of a survey on shower habits conducted in Andalusia, a region in southern Spain facing water scarcity. The interest is to analyse the inequality in time spent showering, creating subpopulations, with sizes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mrow><mn>38</mn><mo>,</mo><mn>74</mn></mrow><mo fence="false" stretchy="false">}</mo></math> </ephtml> , using the number of inhabitants at provincial level.</p> <p>The bias corrected estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> is based on a continuous probabilistic distribution, and for this reason the Kolmogorov-Smirnov (<emph>KS</emph>) Goodness of Fit test is used to fit distributions to the various subpopulations used in this study. From Table 1 we observe that the Lognormal, Fisk, Dagum and Weibull distributions yield <emph>KS p</emph>-values above the usual significance level (5%), and the null hypothesis that data come from the corresponding continuous probabilistic distribution is not rejected. For the various subpopulations, the simulation-based criterion described in the sixth section indicates that the use of a bias correction procedure is recommended, since non-negligible biases and <emph>BR</emph>s >0.1 are expected according to Figure 8. In particular, the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> is expected to underestimate the true Gini index, with higher estimates expected from the bias corrected estimators.</p> <p>For the subpopulation with size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>51</mn></math> </ephtml> derived from ES-SILC, we observe that estimates of the coefficient of skewness and the Gini index are, respectively, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo>=</mo><mn>3.51</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>=</mo><mn>0.476</mn></math> </ephtml> . These results indicate that moderate biases are expected according to Figure 8. As we expected, bias corrected estimators provide higher estimates than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , with values as much as 4.8% larger than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>=</mo><mn>0.476</mn></math> </ephtml> (see the estimation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup></math> </ephtml> based on the Dagum distribution). For <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>503</mn></math> </ephtml> in the WBES population, the estimates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>γ</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi></msub><mo>=</mo><mn>15.85</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>=</mo><mn>0.734</mn></math> </ephtml> indicate the presence of serious biases, and the difference with respect to the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> goes from 2.5% ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>B</mi><mi>p</mi></mrow></msubsup><mo>=</mo><mn>0.752</mn></math> </ephtml> ) to 6.8% ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mrow><mi>c</mi><mo>.</mo><mi>J</mi><mi>o</mi></mrow></msubsup><mo>=</mo><mn>0.784</mn></math> </ephtml> ). For the various subpopulations in this study, we observe that estimates derived from the bias correction procedures are larger than estimates based on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> , a result which coincides with the findings of the third and the fifth sections.</p> <hd id="AN0182244234-15">Discussion</hd> <p>The Gini index is a very popular indicator to measure inequality that has been used in many economic studies. For discrete distributions, the Gini index is usually estimated using a plug-in formulation of a given theoretical definition of the Gini index for continuous distributions. This methodology may introduce a serious bias in comparison to the true (asymptotic) value of the Gini index. Note that the Gini index can also be estimated using techniques such as empirical likelihood ([<reflink idref="bib39" id="ref139">39</reflink>]), but there is no simple application of this method to complex sampling designs. The analysis of alternative estimation methodologies is beyond the scope of this paper, that is, we assume the classical formulations derived from theoretical definitions of the Gini index.</p> <p>First, this article attempts to provide a better overview of the problem of estimating the Gini index by regrouping and classifying the most common empirical versions proposed for discrete distributions, and defined under the two existing statistical theories (infinite and finite populations). Second, this article identifies the scenarios where the bias may be a serious issue, and such scenarios are based on common continuous distributions often used in the modelling of income distributions. For instance, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> (denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>G</mi><mi>N</mi><mi>a</mi></msubsup></math> </ephtml> in finite populations) yields large biases when the Gini index and the sample size are small, but this bias problem can be easily solved by using the midpoint distribution function in the definition of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>a</mi></msubsup></math> </ephtml> . When all the sample observations are different, another solution is to use one of the transformations described in equations (<reflink idref="bib8" id="ref140">8</reflink>) and (<reflink idref="bib9" id="ref141">9</reflink>). In addition, results derived from this study indicate that the various empirical versions of <emph>G</emph> produce serious biases in the presence of heavy-tailed distributions and large Gini indices. Accordingly, bias correction procedures are suggested to mitigate this bias problem, and they are investigated using Monte Carlo simulation studies. We also describe a simulation-based criterion for deciding when to use bias corrections. Finally, bias corrected procedures are illustrated by application to the problem of estimating the Gini index in various real data sets.</p> <p>The empirical bootstrap obtains less biased estimates than alternative bias correction procedures. With infinite populations, the traditional jackknife performs well in terms of relative bias. For finite populations, the rescaled bootstrap may reduce the bias of the existing empirical versions of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> . It is important to note that the empirical bootstrap is a parametric procedure that requires generating sets of data from the probabilistic distribution fitted to the original sample. However, the use of continuous distributions in the modelling of income distributions is a common practice in many real-world applications, and the empirical bootstrap can thus be implemented if this is the case. In addition, it should be noted that for the sake of simplicity the empirical bootstrap bias correction is based only on standard regression functions, but alternative bias functions can also be used, and they may potentially improve the performance of this method. Finally, the empirical bootstrap is more computationally intensive than alternative procedures, but this is not a problem with current computing facilities.</p> <p>The outcome of the grading scale described in the sixth section can help empirical researchers decide whether the specific data they are analysing have non-negligible biases and large bias ratios, meaning the use of bias corrections would therefore be recommended. For heavy-tailed distributions, non-negligible biases may appear in small samples (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>50</mn></math> </ephtml> ) from low estimates of the Gini index (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup><mo>≥</mo><mn>0.2</mn></math> </ephtml> ). For samples with sizes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>200</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1000</mn></math> </ephtml> , non-negligible biases can be expected for estimates of the Gini index >0.4 and 0.5, respectively. Severe biases are not expected when the sample size is larger than 1000. Figure 8 gives a more precise understanding of the conditions required in practice to apply a bias correction, which depend on the sample size and estimates of both the coefficient of skewness and the Gini index.</p> <p>Both bias and MSE measures are important to evaluate the quality of estimators. Numerous authors indicate that the use of bias correction procedures may have an impact on the efficiency of bias corrected estimators. This issue has also been investigated in this article, with the results indicating that said impact is not relevant, especially as the sample size increases. The empirical bootstrap is more efficient than alternative bias correction procedures, but slightly less efficient than the customary empirical versions of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> , and may even have the smallest <emph>MSE</emph>s for large Gini indices. Conventional advice in the literature is to avoid estimators that are considerably biased, so empirical researchers should seek estimators with smaller biases, and then choose one with a small variance. Following this idea, the empirical bootstrap can be good choice for estimating the Gini index in the scenarios discussed in the fifth and the sixth sections. However, alternative bias correction procedures also perform well in terms of bias and efficiency in many situations, and they may be preferable in terms of simplicity.</p> <p>For less skewed distributions (e.g., Weibull and Gamma), the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> is not a problem, and the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>b</mi></msubsup></math> </ephtml> lies within a reasonable range. This implies that bias correction procedures are not required for less skewed distributions. Bias corrections are applied to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> because it shows the best performance in this study. However, such procedures can easily be applied to any other estimation method in the literature.</p> <p>The observed biases may have an important impact on the coverage rates of confidence intervals of the Gini index, especially in the case of the large bias ratios obtained by the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><mi>n</mi><mi>c</mi></msubsup></math> </ephtml> . This implies that bias corrected estimators are highly recommended for the construction of confidence intervals, since they can be invalid and/or undesirable coverage probabilities can be obtained in the case of moderate or severe biases. Large biases are also observed by the bias corrected estimators in the case of large Gini indices and highly skewed distributions. These arguments represent promising directions for future research. For instance, the interval estimation based on bias corrected estimators can be investigated to analyse when such confidence intervals have desirable empirical coverages. Alternative estimation methodologies can also be used to improve the estimation of the Gini index. In particular, it would be interesting to reduce the biases that still remain in the aforementioned extreme situations (highly skewed distributions with large Gini indices). For instance, information from auxiliary variables can be incorporated at the estimation stage, and more accurate results are expected.</p> <hd id="AN0182244234-16">Supplemental Material</hd> <p>Graph: Supplemental material, sj-docx-1-smr-10.1177_00491241231176847 for Exploring and Correcting the Bias in the Estimation of the Gini Measure of Inequality by Juan F. Muñoz, Pablo J. Moya-Fernández and Encarnación Álvarez-Verdejo in Sociological Methods & Research</p> <hd id="AN0182244234-17">Acknowledgments</hd> <p>We thank José M. Pavía for valuable suggestions on an early draft of this article. We thank F. Vendrell-Herrero and F.J. González for providing data sets for this article. We also want to thank the editor and three anonymous reviewers for their valuable comments and suggestions. We thank Y.G. 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In addition, data sets generated from the probabilistic distributions can be obtained using such R scripts. The various real data sets used in this article can be loaded using the file "<emph>Datasets.RData</emph>".</bibtext> </blist> <blist> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research in this article has been supported by the Regional Government of Andalusia and the European Regional Development Fund (project P18-RT-576) and two grants of the University of Granada (Unidad Científica de Excelencia "Desigualdad, Derechos Humanos y Sostenibilidad – DEHUSO" del Plan Propio; and Programa de Ayudas a la revisión de textos científicos de la Facultad de Ciencias Económicas y Empresariales).</bibtext> </blist> <blist> <bibtext> Juan F. Muñoz https://orcid.org/0000-0001-7427-6630 Pablo J. Moya-Fernández https://orcid.org/0000-0003-0980-3849 Encarnación Álvarezhttps://orcid.org/0000-0002-0473-6037</bibtext> </blist> <blist> <bibtext> Supplemental material for this article is available online.</bibtext> </blist> </ref> <aug> <p>By Juan F. Muñoz; Pablo J. Moya-Fernández and Encarnación Álvarez-Verdejo</p> <p>Reported by Author; Author; Author</p> <p></p> <p>Juan F. Muñoz is a full professor at the department of Quantitative Methods in Economics and Business at the University of Granada. He is especially interested in the estimation of parameters related to economics, business, and other topics included in social sciences, and especially on poverty and inequality measures. Some of the more recent publications appeared in Social Indicators Research, Journal of Official Statistics, Journal of Applied Statistics, Total Quality Management & Business Excellence, Quality and Reliability Engineering International, Total Quality Management, International Journal of Operations & Production Management, and Communications in Statistics - Simulation and Computation.</p> <p>Pablo J. Moya-Fernández is an associate professor at the department of Quantitative Methods in Economics and Business at the University of Granada. His research deals with quantitative methods applied to environmental problems and parameter estimation applied to economic and business issues as measures of poverty and inequality. Some of his more recent publications appeared in Energy Economics, Sustainable Production and Consumption, International Journal of Water Resources Development, Communications in Statistics-Simulation and Computation and Journal of the Knowledge Economy.</p> <p>Encarnación Álvarez-Verdejo is an associate professor at the department of Quantitative Methods in Economics and Business at the University of Granada. She is especially interested in the use of statistical methods for the estimation of poverty and inequality measures, and other parameters in social sciences. Some of the more recent publications appeared in Social Indicators Research, Journal of Applied Statistics, Quality and Reliability Engineering International, and Communications in Statistics - Simulation and Computation.</p> </aug> <nolink nlid="nl1" bibid="bib33" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib22" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib42" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib50" firstref="ref4"></nolink> <nolink nlid="nl5" bibid="bib49" firstref="ref6"></nolink> <nolink nlid="nl6" bibid="bib18" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib55" firstref="ref8"></nolink> <nolink nlid="nl8" bibid="bib53" firstref="ref10"></nolink> <nolink nlid="nl9" bibid="bib30" firstref="ref11"></nolink> <nolink nlid="nl10" bibid="bib32" firstref="ref12"></nolink> <nolink nlid="nl11" bibid="bib27" firstref="ref13"></nolink> <nolink nlid="nl12" bibid="bib11" firstref="ref14"></nolink> <nolink nlid="nl13" bibid="bib14" firstref="ref15"></nolink> <nolink nlid="nl14" bibid="bib29" firstref="ref16"></nolink> <nolink nlid="nl15" bibid="bib26" firstref="ref17"></nolink> <nolink nlid="nl16" bibid="bib15" firstref="ref18"></nolink> <nolink nlid="nl17" bibid="bib47" firstref="ref20"></nolink> <nolink nlid="nl18" bibid="bib40" firstref="ref21"></nolink> <nolink nlid="nl19" bibid="bib46" firstref="ref24"></nolink> <nolink nlid="nl20" bibid="bib12" firstref="ref25"></nolink> <nolink nlid="nl21" bibid="bib54" firstref="ref26"></nolink> <nolink nlid="nl22" bibid="bib20" firstref="ref27"></nolink> <nolink nlid="nl23" bibid="bib43" firstref="ref28"></nolink> <nolink nlid="nl24" bibid="bib60" firstref="ref31"></nolink> <nolink nlid="nl25" bibid="bib23" firstref="ref34"></nolink> <nolink nlid="nl26" bibid="bib48" firstref="ref40"></nolink> <nolink nlid="nl27" bibid="bib38" firstref="ref41"></nolink> <nolink nlid="nl28" bibid="bib56" firstref="ref46"></nolink> <nolink nlid="nl29" bibid="bib21" firstref="ref48"></nolink> <nolink nlid="nl30" bibid="bib10" firstref="ref61"></nolink> <nolink nlid="nl31" bibid="bib13" firstref="ref70"></nolink> <nolink nlid="nl32" bibid="bib17" firstref="ref74"></nolink> <nolink nlid="nl33" bibid="bib19" firstref="ref78"></nolink> <nolink nlid="nl34" bibid="bib57" firstref="ref79"></nolink> <nolink nlid="nl35" bibid="bib41" firstref="ref80"></nolink> <nolink nlid="nl36" bibid="bib28" firstref="ref81"></nolink> <nolink nlid="nl37" bibid="bib34" firstref="ref82"></nolink> <nolink nlid="nl38" bibid="bib31" firstref="ref83"></nolink> <nolink nlid="nl39" bibid="bib51" firstref="ref88"></nolink> <nolink nlid="nl40" bibid="bib25" firstref="ref95"></nolink> <nolink nlid="nl41" bibid="bib16" firstref="ref104"></nolink> <nolink nlid="nl42" bibid="bib44" firstref="ref115"></nolink> <nolink nlid="nl43" bibid="bib45" firstref="ref116"></nolink> <nolink nlid="nl44" bibid="bib35" firstref="ref118"></nolink> <nolink nlid="nl45" bibid="bib36" firstref="ref119"></nolink> <nolink nlid="nl46" bibid="bib59" firstref="ref120"></nolink> <nolink nlid="nl47" bibid="bib58" firstref="ref132"></nolink> <nolink nlid="nl48" bibid="bib24" firstref="ref137"></nolink> <nolink nlid="nl49" bibid="bib52" firstref="ref138"></nolink> <nolink nlid="nl50" bibid="bib39" firstref="ref139"></nolink>
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PubTypeId: academicJournal
PreciseRelevancyScore: 0
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Exploring and Correcting the Bias in the Estimation of the Gini Measure of Inequality
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Juan+F%2E+Muñoz%22">Juan F. Muñoz</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-7427-6630">0000-0001-7427-6630</externalLink>)<br /><searchLink fieldCode="AR" term="%22Pablo+J%2E+Moya-Fernández%22">Pablo J. Moya-Fernández</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-0980-3849">0000-0003-0980-3849</externalLink>)<br /><searchLink fieldCode="AR" term="%22Encarnación+Álvarez-Verdejo%22">Encarnación Álvarez-Verdejo</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-0473-6037">0000-0002-0473-6037</externalLink>)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2025 54(1):237-274.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 38
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Evaluative
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Bias%22">Bias</searchLink><br /><searchLink fieldCode="DE" term="%22Educational+Indicators%22">Educational Indicators</searchLink><br /><searchLink fieldCode="DE" term="%22Equal+Education%22">Equal Education</searchLink><br /><searchLink fieldCode="DE" term="%22Monte+Carlo+Methods%22">Monte Carlo Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Sampling%22">Sampling</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Inference%22">Statistical Inference</searchLink><br /><searchLink fieldCode="DE" term="%22Income%22">Income</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/00491241231176847
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0049-1241<br />1552-8294
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The Gini index is probably the most commonly used indicator to measure inequality. For continuous distributions, the Gini index can be computed using several equivalent formulations. However, this is not the case with discrete distributions, where controversy remains regarding the expression to be used to estimate the Gini index. We attempt to bring a better understanding of the underlying problem by regrouping and classifying the most common estimators of the Gini index proposed in both infinite and finite populations, and focusing on the biases. We use Monte Carlo simulation studies to analyse the bias of the various estimators under a wide range of scenarios. Extremely large biases are observed in heavy-tailed distributions with high Gini indices, and bias corrections are recommended in this situation. We propose the use of some (new and traditional) bootstrap-based and jackknife-based strategies to mitigate this bias problem. Results are based on continuous distributions often used in the modelling of income distributions. We describe a simulation-based criterion for deciding when to use bias corrections. Various real data sets are used to illustrate the practical application of the suggested bias corrected procedures.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1457727
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1457727
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1177/00491241231176847
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 38
        StartPage: 237
    Subjects:
      – SubjectFull: Bias
        Type: general
      – SubjectFull: Educational Indicators
        Type: general
      – SubjectFull: Equal Education
        Type: general
      – SubjectFull: Monte Carlo Methods
        Type: general
      – SubjectFull: Sampling
        Type: general
      – SubjectFull: Statistical Inference
        Type: general
      – SubjectFull: Income
        Type: general
    Titles:
      – TitleFull: Exploring and Correcting the Bias in the Estimation of the Gini Measure of Inequality
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Juan F. Muñoz
      – PersonEntity:
          Name:
            NameFull: Pablo J. Moya-Fernández
      – PersonEntity:
          Name:
            NameFull: Encarnación Álvarez-Verdejo
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 02
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 0049-1241
            – Type: issn-electronic
              Value: 1552-8294
          Numbering:
            – Type: volume
              Value: 54
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: Sociological Methods & Research
              Type: main
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