Improving the ARL profile and the accuracy of its calculation for Poisson EWMA charts

Abstract

The Poisson exponentially weighted moving average (PEWMA) chart was proposed in 1998 by Borror, Champ, and Rigdon to monitor the mean of counts of nonconformities. This chart regrettably fails to have an in‐control average run length (ARL) larger than any out‐of‐control ARL, ie, the PEWMA chart is ARL‐biased. Moreover, due to the discrete character of its control statistic the PEWMA chart and the resulting subtleties of its ARL calculation, it is difficult to set the control limits in such way that the in‐control ARL takes a desired value, say ARL0. In this paper, we establish an improved Markov chain technique to approximate the ARL of EWMA for count output; propose two ARL‐unbiased counterparts of the PEWMA chart; and use the R statistical software to provide illustrations of these charts with a decidedly superior ARL profile and an in‐control ARL equal to ARL0. We also compare the ARL performance of the proposed charts with the one of a few competing control charts for the mean of independent and identically distributed (i.i.d.) Poisson counts.

Design of attribute EWMA type control charts with reliable run length performance

“…Table 4: Optimal combinations (λ * , K * ) and corresponding out-of-control ARL values for p 0 ∈ {0.05, 0.1, 0.15, 0.2}, τ ∈ {1.1, 1.2, 1.5, 2} and σ ∈ {0.1, 0.125, 0.15, 0.2} for the upper-sided BEWMA chart with "continuousify" in an efficient way. Finally, a comparison with the recent "Splitting Markov Chain" approach proposed by Morais and Knoth (2020) has shown that our "continuousify" method performs in a similar way in the case of the PEWMA control chart.…”

“…It is interesting to note that this topic (efficient ARL computation for EWMA control charts monitoring discrete data) has also grasped the attention of other researchers and, very recently, Morais and Knoth (2020) have proposed an alternative method called the "Splitting Markov Chain". This approach has been applied to a two-sided PEWMA control chart (while, in this paper, we investigate a max-type upper sided PEWMA control chart).…”

“…Therefore, in an attempt to extend the preliminary work by Wu et al (2020), the goal of this paper is i) to propose a method, called "continuousify", that allows to compute the RL properties of any attribute EWMA type control charts in a reliable way, ii) to demonstrate the use of this method in the case of the Poisson and binomial distributions, iii) to design the corresponding PEWMA (for Poisson EWMA) and BEWMA (for Binomial EWMA) and iv) to compare the results with the ones obtained through Monte Carlo simulations as well as with the ones presented in the very recent contribution of Morais and Knoth (2020) in which the authors tackle the same problem (for the PEWMA control chart only) but in a different way.…”

An unbiased function‐based Poisson adaptive EWMA control chart for monitoring range of shifts

“…15 Abbasi 16 developed Poisson progressive mean (PPM) charts under the zero and steady-state situations. The interested readers are referred to Rakitzis et al, 17 Morais et al, 18 Cabral Morais and Knoth, 19 Abbas et al, 20 and Aly et al 21 for more literature on Poisson charting schemes.…”

Enhancing Control: Unveiling the Performance of Poisson EWMA Charts through Simulation with Poisson Mixture DATA

“…A new PEWMA chart is introduced by Sukparungsee [18] improved based on square transformation of Poisson data. In considering PEWMA as ARL-biased, Morais and Konth [19] used the Markov chain to determine ARL accurately and compared some competing results of previous studies. Although the Poisson distribution is frequently used to describe count data, it assumes that variance is equal to the mean (equi dispersion), which could lead to incorrect inferences in the case of overdispersion [20].…”