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 ine↵ective. 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 o↵ered.