A robust CUSUM scheme with a weighted likelihood ratio to monitor an overdispersed counting process

The Poisson cumulative sum (CUSUM) control chart is a method to monitor count data, which are commonly modeled by a Poisson distribution. However, the assumption of a Poisson distribution may not be valid in practice and the robustness of the Poisson CUSUM chart is shown to be poor when the volatility is greater than expected, a phenomenon known as overdispersion. Motivated by this, we propose an improved weighted Poisson CUSUM control chart based on the weighted log-likelihood ratio. For the purpose of discounting the unexplained volatility, smaller weights are assigned to the larger shifts. The continuity, monotonicity and convergence of the cumulative statistic proved in this paper make the design more practical. Compared with the conventional Poisson CUSUM and the Winsorized Poisson CUSUM control charts, the new weighted Poisson CUSUM scheme performs most robustly under overdispersion. In addition, the fast initial response (FIR) feature makes the chart more sensitive to start-up cases. A real-data example of monitoring daily orders on a start-up E-commerce company illustrates the superiorities of the proposed scheme.