The Poisson INAR(1) CUSUM chart under overdispersion and estimation error

Abstract

The Poisson INAR(1) CUSUM chart has been proposed to monitor integer-valued autoregressive processes of order 1 with Poisson marginals. The effectiveness of this chart has been shown under the assumptions of Poisson marginals and known in-control process parameters, but these assumptions may not be very well satisfied in practical applications. This article investigates the practical issues concerning applications of the Poisson INAR(1) CUSUM chart, considering average run lengths obtained through a bivariate Markov chain approach. First, the effects of deviations from the assumed Poisson model are investigated when there is overdispersion. Design recommendations for achieving robustness are provided along with an extension, the Winsorized Poisson INAR(1) CUSUM chart. Next, analyzing the conditional average run length performance under some hypothetical cases of parameter estimation, it is shown that estimation errors may severely affect the chart’s performance. The marginal average run length performance is used to derive sample size recommendations. An example for monitoring the number of beds occupied at a hospital emergency department is used to illustrate the proposed approach.

Keywords:

• Attributes control chart
• Poisson distribution
• autocorrelation
• INAR(1) model
• CUSUM chart
• Winsorization

Acknowledgement

The authors would like to thank the referees for carefully reading the paper and for their comments, which greatly improved the article.