Bayesian Monitoring of Times Between Events: The Shewhart tr-Chart

Abstract

The traditional (frequentist) tr-chart is a Shewhart-type chart useful for monitoring times between events (interarrival times) following an exponential distribution. This problem often arises in high-yield processes where the defect rate is low and hence the conventional attribute charts such as the c-chart and the u-chart are often ineffective. We consider this problem under the Bayesian framework and propose a Bayesian tr-chart when the exponential rate parameter is unknown. The Bayesian tr-chart is a Shewhart-type chart that incorporates parameter uncertainty via a prior and a posterior distribution, unlike the traditional tr-chart. The control limits are constructed from the predictive distribution of a plotting statistic. The performance of the proposed chart is evaluated and comparisons are made with the traditional tr-chart. The Bayesian chart is seen to be advantageous in certain situations. An illustrative example is given and a summary and conclusions are offered.