Self-Starting Monitoring Scheme for Poisson Count Data With Varying Population Sizes

Abstract

In this paper we consider the problem of monitoring Poisson rates when the population sizes are time-varying and the nominal value of the process parameter is unavailable. Almost all previous control schemes for the detection of increases in the Poisson rate in Phase II are constructed based on assumed knowledge of the process parameters, e.g., the expectation of the count of a rare event when the process of interest is in control. In practice, however, this parameter is usually unknown and not able to be estimated with a sufficiently large number of reference samples. A self-starting EWMA control scheme based on a parametric bootstrap method is proposed. The success of the proposed method lies in the use of probability control limits, which are determined based on the observations during rather than before monitoring.Simulation studies show that our proposed scheme has good in-control and out-of-control performance under various situations. In particular, our proposed scheme is useful in rare event studies during the start-up stage of a monitoring process.

A predictive Bayesian approach to sequential time‐between‐events monitoring

“…Some more recent literature in this direction can be seen in the previous studies . Moreover, we also refer to Shen et al, Keefe et al, and references cited therein for some recent contributions in self‐starting with the frequentist approach.…”

“…Since the updated control limits of each time point are determined sequentially by incorporating the in‐control observation to update the posterior and the predictive distributions, we believe that monitoring can be activated at the startup. It is worth mentioning that one may consider our sequential approach similar to some existing frequentist approaches, like the change point problem presented in Shen et al that also requires a small phase‐I data set to initiate the process monitoring. Although the approach of Shen et al is self‐adaptive, we point out that the current approach is fundamentally different because of the formal mechanism of Bayesian predictive distribution that allows to incorporate prior information in the analysis.…”

“…Thus, to construct a predictive type control chart, it is crucial to find the control limits so that the chart has satisfactory run length. A chart is usually unacceptable if the in‐control ARL is obtained with higher probability of false alarms or with short run‐length as compared with a geometric distribution . Hence, we require the control limits such that the conditional probability that the charting statistic crosses the control limit at present, given that there was no false alarm before the current time point, is a prefixed constant.…”

Self-starting single control charts for multivariate processes: a comparison of methods

“…'Self-starting methods' were proposed to amalgamate phase I and phase II studies. These methods involve sequentially updating the parameters with every new successive process reading and monitoring the process to see whether it is in-control or not (Hawkins, 1987;Hawkins & Maboudou-Tchao, 2007;Keefe et al, 2015;Khosravi & Amiri, 2019;Quesenberry, 1991;Shen et al, 2016). The unknown parameter change point estimation procedures (Hawkins & Zamba, 2005;Zamba & Hawkins, 2009) can also be classified as self-starting methods.…”

New Bayesian methods for quality control applications

“…Quesenberry (1991aQuesenberry ( , 1995b) built shewhart-type charts based on normal approximations for Poisson and Binomial data, and the monitoring performance can be improved if CUSUM and EWMA setups are employed. Shen et al (2015) proposed to inspect possible mean shifts of Poisson data by heuristic tests. But the method requires historical data to initialize the monitoring.…”