In-control performance of the joint Phase II − S control charts when parameters are estimated

ABSTRACT The issue of the effects of parameter estimation on the in-control performance of control charts has motivated researchers for several decades. In this context, recently, acknowledging what has been called by some the practitioner-to-practitioner variability, a new perspective has been advocated, namely, the study of the conditional distribution of the in-control average run length (or the conditional false-alarm rate), which is more meaningful in practice. Adopting this new perspective, some authors have analyzed the conditional distribution of the false-alarm rate (or of the in-control average run length) of and of S charts separately. However, since the and S charts are not typically used separately but together or jointly in many applications, here we study the effects of parameter estimation on the performance of the two charts applied jointly (called the joint charts). For the joint charts, defining the joint false-alarm rate as the probability that at least one of the two charts ( and S) issues a false alarm, we obtain its conditional distribution, some quantiles of interest (upper prediction bounds for it) and the number of Phase I samples required to guarantee that with a high probability the conditional joint false alarm rate will not exceed a maximum tolerated value. We assume normality, consider Sp (the square root of the pooled variance) as the Phase I estimator of the process standard deviation, and consider two possibilities regarding the chart: (1) centered at and (2) centered at a specified target value. The results show (and we formally prove) that, whereas the required number of Phase I samples may be very large for the joint charts, interestingly, it lies between the corresponding numbers of samples required by the chart and by the S chart individually; so, considering the performance of the charts from the perspective of their joint use may slightly alleviate the required number of Phase I samples.

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