Exact Run Length Evaluation on Extended EWMA Control Chart for Autoregressive Process

Abstract

Extended Exponentially Weighted Moving Average (Extended EWMA or EEWMA) control chart is one of the control charts which can quickly detect a small shift. The average run length (ARL) measures the performance of control chart. Due to the derivation of the explicit formulas for ARL on the EEWMA control chart for the autoregressive AR(p) process has not previously been reported. The aim of the article is to derive explicit formulas of ARL using a Fredholm integral equation of the second kind on EEWMA control chart for Autoregressive process, as AR(2) and AR(3) processes with exponential white noise. The accuracy of the solution obtained with the EEWMA control chart was compared to the numerical integral equation (NIE) method and extended to compare performance with Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) control charts. The results show that the ARL obtained by the explicit formula and the NIE method are hardly different but ARL of explicit formula is less the computational (CPU) time than ARL of NIE method. The performance of EEWMA control chart is better than the CUSUM and EWMA control charts for all situations except when the large shift sizes the EEWMA control chart performed as well as the EWMA control chart for AR(2) and AR(3) processes. And then, the EEWMA control chart is also extended to compare efficiency of EEWMA control chart with various λ. An exponential smoothing parameter of 0.05 is recommended. In addition, the simulation study, and efficacy illustration with real data on new COVID-19 cases in Thailand and Vietnam provided similar results.