An ARL-unbiased modified chart for monitoring autoregressive counts with geometric marginal distributions

Geometrically distributed counts arise in the industry. Ideally, they should be monitored using a control chart whose average run length (ARL) function achieves a maximum when the process is in control; that is, the chart is ARL-unbiased. Moreover, its in-control ARL should coincide with a reasonably large and prespecified value. Because dependence among successive geometric counts is occasionally a more sensible assumption than independence, we assess the impact of using an ARL-unbiased chart specifically designed for monitoring independent geometric counts when, in fact, these counts are autocorrelated. We derive an ARL-unbiased modified chart for monitoring geometric first-order integer-valued autoregressive or GINAR(1) counts. We provide compelling illustrations of this chart and discuss its use to monitor other autoregressive counts with a geometric marginal distribution.