An adaptive EWMA control chart based on Hampel function to monitor the process location parameter

Abstract

The classical memory Exponential Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) control charts (CCs) are important to detect smallto-moderate sizes of shift in the process location and/or dispersion parameters. To make the memory control chart (CC) more efficient for broad range of shift, the score functions in particular Huber and Bi-square functions are incorporated in the classical memory CCs. These score functions help to formulate the parameters of memory CCs and are called Adaptive EWMA (AEWMA) and Adaptive CUSUM (ACUSUM). The AEWMA and ACUSUM are capable to handle various sizes (i.e., small, moderate, and large) of shift. The aim of this study is to utilize Hampel function as the alternative of the mentioned score functions, and propose a new AEWMA CC, symbolized as AEWMA Hample . To develop the proposed AEWMA Hample CC, the standard EWMA statistic and Hampel function are incorporated in the classical EWMA CC structure. The proposed AEWMA Hample CC is also efficient to handle certain sizes of shift in the process location parameter. Performance evaluation procedures, like average run length is considered for a particular shift. Similarly, the performance comparison index, relative average run length, and extra quadratic loss function are estimated over a range of shift. All these performance measures are calculated from run length (RL) characteristic of the proposed control chart. An algorithm in MATLAB is developed based on Monte Carlo simulation procedure to generate the RLs. Some existing control charts including EWMA, CUSUM, MEC, MCE, ACUSUM (1) c , ACUSUM (2) c , IACCUSUM, AEWMA ∅ hu , and AEWMA ∅ bs are taken for comparison reason.For specific sizes of shift, the proposed AEWMA Hample CC reveals the superiority. Besides, to address the practical significance for practitioners and quality engineers, the real-life and simulated data examples are considered.

Corrections to an adaptive EWMA control chart based on Hampel function to monitor the process location parameter by Zaman et al. (2023)

“…The study intention is to identify and rectify the errors in the article of Zaman et al 1 . Some errors have been corrected, and the results have been reproduced by considering the corrections.…”

“…Zaman et al 1 . proposed an adaptive exponential weighted moving average (AEWMA), denoted as an AEWMA Hampel control chart (CC).…”

“…are actually missed interpreted based on the mentioned text and information. For example, in Section 3 of Zaman et al., 1 $${e}_i = {x}_i\ - EWM{A}_{( {i - 1} )}$$, $$LC{L}_{hampel( i )} = \ \mu - {L}_{hampel}\sigma \sqrt {\frac{\vartheta }{{2 - \vartheta }}[ {1 - {{( {1 - \vartheta } )}}^2} ]} $$, and $$UC{L}_{hampel( i )} = \ \mu + {L}_{hampel}\sigma \sqrt {\frac{\vartheta }{{2 - \vartheta }}[ {1 - {{( {1 - \vartheta } )}}^2} ]} $$ should be replaced by the $${e}_i = {x}_i\ - AEWM{A}_{H( {i - 1} )}$$, $$LC{L}_{hampel( i )} = \ \mu - {L}_{hampel}\sigma \sqrt {\frac{\vartheta }{{2 - \vartheta }}[ {1 - {{( {1 - \vartheta } )}}^{2i}} ]} $$, and …”

“…In contrary, the true expressions are $${e}_i = {x}_i\ - AEWM{A}_{H( {i - 1} )}$$, $$LC{L}_{hampel( i )} = \ \mu - {L}_{hampel}\sigma \sqrt {\frac{\vartheta }{{2 - \vartheta }}[ {1 - {{( {1 - \vartheta } )}}^{2i}} ]} $$, and $$UC{L}_{hampel( i )} = \ \mu + {L}_{hampel}\sigma \sqrt {\frac{\vartheta }{{2 - \vartheta }}[ {1 - {{( {1 - \vartheta } )}}^{2i}} ]} $$ as motivated by Capizzi and Masarotto 2 and Montgomery 3 . By utilizing true expressions and using Monte Carlo simulation, the Zaman et al 1 . study results are re‐produced with the same parameters and are given in Table 1.…”

“…In contrary, the true expressions are 𝑒 𝑖 = 𝑥 𝑖 − 𝐴𝐸𝑊𝑀𝐴 𝐻(𝑖−1) , 𝐿𝐶𝐿 ℎ𝑎𝑚𝑝𝑒𝑙(𝑖) = 𝜇 − 𝐿 ℎ𝑎𝑚𝑝𝑒𝑙 𝜎 motivated by Capizzi and Masarotto 2 and Montgomery 3. By utilizing true expressions and using Monte Carlo simulation, the Zaman et al1 study results are re-produced with the same parameters and are given in…”