Joint monitoring of mean and dispersion of count data

Abstract

Count data are often described by the Poisson distribution, which requires identical mean and variance, namely equi-dispersion. However, in practical situations, count data usually exhibit either over-dispersion with variance larger than mean, or under-dispersion with variance smaller than mean. Therefore, traditional approaches that focus on only mean shifts, such as the c-chart, cannot monitor count data with over/under-dispersion efficiently. To monitor mean and dispersion of count data simultaneously, this paper adopts Conway–Maxwell–Poisson (COM–Poisson) distributions to fit count data with over/under-dispersion, and constructs a control chart based on the likelihood ratio test. The proposed chart is powerful in detecting both mean and dispersion shifts of count data with either over-dispersion or under-dispersion. Numerical simulations have demonstrated its performance in various cases.