An ARL‐unbiased c‐chart

Abstract

In statistical process control (SPC), it is usual to assume that counts have a Poisson distribution. The non-negative, discrete, and asymmetrical character of a control statistic with such a distribution and the value of its target mean may prevent the quality control practitioner to deal with a c-chart with a pre-specified in-control average run length (ARL) or the ability to control not only increases but also decreases in the mean of those counts in a timely fashion. Furthermore, the c-charts proposed in the SPC literature tend to be ARL-biased, in the sense that some out-of-control ARL values are larger than the in-control ARL. In this paper, we explore the notions of randomized and uniformly most powerful unbiased tests to eliminate the bias of the ARL function of the c-chart.

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

“…For a glimpse of the existing work on ARL‐unbiased designs involving absolutely continuous and discrete control statistics, the reader is referred to the brief reviews in Knoth and Morais and Morais and Knoth . They lead to the conclusion that the SPC literature on ARL‐unbiased charts depending on discrete control statistics is not abundant: Paulino et al and Morais et al are, as far as we have investigated, three exceptions when it comes to the monitoring of the Poisson mean.…”

“…The main goal of Section is to briefly compare the performance of the two ARL‐unbiased PEWMA charts we proposed with two other charts for i.i.d. Poisson counts: the PEWMA chart and the ARL‐unbiased ‐chart …”

“…The control limits and randomization probabilities of the ARL‐unbiased ‐chart (unb. c‐chart) were obtained using the algorithm proposed by Paulino et al, considering the false alarm rate equal to .…”

“…Poisson counts: the PEWMA chart and the ARL-unbiased c-chart. 16 Table 5 comprises the control limits and other relevant parameters of the contending charts we mentioned above for 0 = 0.5, 1, 2, 4, 8, 16 and ARL 0 = 500; the smoothing constant, number of transient states, and the grid resolution used for all PEWMA charts were set to λ = 0.1, N = 501, and 10 −6 (respectively). The control limits and randomization probabilities of the ARL-unbiased c-chart (unb.…”

“…The control limits and randomization probabilities of the ARL-unbiased c-chart (unb. c-chart) were obtained using the algorithm proposed by Paulino et al, 16 considering the false alarm rate equal to 1 ARL 0 . It is apparent from Table 5 that the critical constants and control limits of the ARL-unbiased PEWMA charts with and without randomization coincide down to five decimal places.…”