Change point estimation for monotonically changing Poisson rates in SPC

Abstract

Knowing when a process has changed would simplify the search for and identification of the special cause. Although several change point methods have been suggested, many of them rely on the assumption that the effect present in the process output follows some known form (e.g. sudden shift or linear trend). Since processes are often influenced by several input factors, sudden shifts and linear trends do not always adequately describe the true nature of the process behavior. In this paper, we propose a maximum likelihood estimator for the change point of a Poisson rate parameter without requiring exact a priori knowledge regarding the form of the effect present. Instead, we assume the form of effect present can be characterized as belonging to the set of monotonic effects. We compare the proposed change point estimator to the commonly used maximum likelihood estimator for the process change point derived under a sudden and persistent shift assumption. We do this for a number of monotonic effects and following a signal from a Poisson CUSUM control chart. We conclude that it is better to use the proposed change point estimator when the form of the effect present is only known to be monotonic. The results show that the proposed estimator provides process engineers with an accurate and useful estimate of the last observation obtained from the unchanged process regardless of the form of monotonic effect that may be present.