On ARL-unbiased c-charts for INAR(1) Poisson counts

Counts of nonconformities are frequently assumed to have a Poisson distribution. The integer and asymmetrical character of this distribution and the value of its target mean may prevent the quality control operator to deal with a chart with a pre-specified in-control average run length (ARL) and the ability to promptly detect both increases and decreases in the mean of those counts. Moreover, as far as we know, the c-chart proposed to monitor the mean of first-order integer-valued autoregressive [INAR(1)] Poisson counts tends to be ARL-biased, in the sense that it takes longer, in average, to detect some shifts in the process mean than to trigger a false alarm. In this paper, we capitalize on the randomization of the emission of a signal and on a nested secant rule search procedure not only to eliminate the bias of the ARL function of the c-chart for the mean of INAR(1) Poisson counts, but also to bring its in-control ARL exactly to a pre-specified and desired value. Striking illustrations of the resulting ARL-unbiased c-chart are provided.