Explicit ARL Derivation of Extended EWMA Control Chart to Detect Changes in the First-Order SAR with Quadratic Trend and Its Application.

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Title:	Explicit ARL Derivation of Extended EWMA Control Chart to Detect Changes in the First-Order SAR with Quadratic Trend and Its Application.
Authors:	Areepong, Yupaporn¹ yupaporn.a@sci.kmutnb.ac.th, Karoon, Kotchaporn² kotchapornk@nu.ac.th
Source:	IAENG International Journal of Applied Mathematics. May2026, Vol. 56 Issue 5, p1924-1935. 12p.
Subjects:	Quality control charts, Autoregressive models, Fredholm equations, Numerical solutions to integral equations, Market volatility, Fixed point theory
Abstract:	The performance of control charts to detect shifts may be evaluated using the basic performance metric of Average Run Length (ARL). This study compares ARL estimates achieved using numerical integral equation (NIE) approaches, such as the Midpoint, Trapezoidal, Simpson's 1/3, Simpson's 3/8, and Boole's rules, with explicit ARL formulations derived for the Extended Exponentially Weighted Moving Average (extended EWMA) control chart. The suggested approach is based on an exponential white noise driven first-order seasonal autoregressive model with a quadratic trend, SAR(1)L model. By means of the Fredholm integral equation, the explicit ARL expressions are obtained, and the solution's uniqueness is rigorously shown by means of Banach's Fixed Point Theorem. The numerical findings show that when compared to NIE-based methods, the explicit ARL formulae significantly reduce computing cost and provide calculations that are almost immediate. The explicit ARL derivation of the extended EWMA had been extended to compare its performance with that of the traditional EWMA chart, and the results showed better performance. Finally, Monthly stock price series of Nike, characterized by substantial volatility driven by market dynamics, are employed to evaluate and confirm the superior performance of the proposed Extended EWMA control chart compared with the conventional EWMA chart. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

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Abstract:	The performance of control charts to detect shifts may be evaluated using the basic performance metric of Average Run Length (ARL). This study compares ARL estimates achieved using numerical integral equation (NIE) approaches, such as the Midpoint, Trapezoidal, Simpson's 1/3, Simpson's 3/8, and Boole's rules, with explicit ARL formulations derived for the Extended Exponentially Weighted Moving Average (extended EWMA) control chart. The suggested approach is based on an exponential white noise driven first-order seasonal autoregressive model with a quadratic trend, SAR(1)L model. By means of the Fredholm integral equation, the explicit ARL expressions are obtained, and the solution's uniqueness is rigorously shown by means of Banach's Fixed Point Theorem. The numerical findings show that when compared to NIE-based methods, the explicit ARL formulae significantly reduce computing cost and provide calculations that are almost immediate. The explicit ARL derivation of the extended EWMA had been extended to compare its performance with that of the traditional EWMA chart, and the results showed better performance. Finally, Monthly stock price series of Nike, characterized by substantial volatility driven by market dynamics, are employed to evaluate and confirm the superior performance of the proposed Extended EWMA control chart compared with the conventional EWMA chart. [ABSTRACT FROM AUTHOR]
ISSN:	19929978