# Probability and Statistics Seminar

### On ARL-unbiased c-charts for i.i.d. and INAR(1) Poisson counts

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 distribution and the value of its target mean may prevent the quality control practitioner to deal with a c-chart with:

1. a positive lower control limit and the ability to control not only increases but also decreases in the mean of those counts in a timely fashion;
2. a pre-specified in-control average run length (ARL).

Furthermore, as far as we have investigated, the c-charts proposed in the SPC literature tend not to be ARL-unbiased. (The term ARL-unbiased is used here to coin any control chart for which all out-of-control ARL values are smaller than  the in-control ARL.)

In this talk, we explore the notions of unbiased, randomized and uniformly most powerful unbiased tests (resp. randomization of the emission of a signal and a nested secant rule search procedure) to:

1. eliminate the bias of the ARL function of the c-chart for the mean of i.i.d. (resp. first-order integer-valued autoregressive, INAR(1)) Poisson counts;
2. bring the in-control ARL exactly to a pre-specified and desired value.

We use the R statistical software to provide striking illustrations of the resulting ARL-unbiased c-charts.

Joint work with Sofia Paulino and Sven Knoth