Progressive mean control chart is not a special case of an exponentially weighted moving average control chart

Abstract

The progressive mean (PM) statistic is based on a simple idea of accumulating information of each subgroup by calculating the average progressively. Its weighting structure is based on a subgroup number that changes arithmetically, which makes the PM chart unique and efficient compared with the existing classical memory control charts. In a recent article (see reference 1), it was claimed that the PM chart is a special case of the exponentially weighted moving average (EMWA) chart. In this article, it is shown that even though the PM statistic can

Progressive mean as a special case of exponentially weighted moving average: Discussion

“…They wrongfully proved that the variances of PM statistics in Equation (2) and Equation (3) are not same and hence deducted that PM chart is not a special case of EWMA chart. This study points out the slipup of Zafar et al 9 and finds the true variance of PM statistic in (3) and proves that it is identical to Var(𝑃𝑀 𝑖 ) in Equation (2).…”

“…Zafar et al 9 . laid out the foundation of their derivation by assuming that the variance of PM statistic in Equation () can be written as $${\sigma ^2}[ {\frac{{( {1/i} )}}{{2 - ( {1/i} )}}\{ {1 - {{( {1 - \frac{1}{i}} )}^{2i}}} \}} ]$$ using the variance expression of EWMA chart in Equation ().…”

“…A recent study published by Zafar et al 9 . tried to disprove the result established by Abbas 1 through some mathematical manipulations of their own.…”

“…A recent study published by Zafar et al 9 tried to disprove the result established by Abbas 1 through some mathematical manipulations of their own. They wrongfully proved that the variances of PM statistics in Equation (2) and Equation (3) are not same and hence deducted that PM chart is not a special case of EWMA chart.…”

“…SLIPUP BY ZAFAR ET AL. 9 Zafar et al 9 laid out the foundation of their derivation by assuming that the variance of PM statistic in Equation ( 2 }] using the variance expression of EWMA chart in Equation (1). It is to be noted here that the variance of EWMA statistic in ( 1) is true only for a static value of 𝜆 and hence it cannot be used when 𝜆 is a function of 𝑖 (i.e., 𝜆 𝑖 = 𝑖 −1 .)…”