The Generalized Likelihood Ratio Chart for Monitoring a Proportion with Autocorrelation

Abstract

When monitoring a proportion p, it is usually assumed that the binary observations are independent. This paper investigates the problem of monitoring p when the binary observations follow a two‐state Markov chain model with first‐order dependence. A Markov binary generalized likelihood ratio (MBGLR) chart based on a likelihood ratio statistic with an upper bound on the estimate of p is proposed. The MBGLR chart is used to monitor a continuous stream of autocorrelated binary observation. The MBGLR chart with a relatively large upper bound has good overall performance over a wide range of shifts. The extra number of defectives is defined to measure the loss when using control charts for monitoring p. The MBGLR chart is optimized over a range of upper bounds for the MLE of p. The numerical results show that the optimized MBGLR chart has a smaller extra number of defectives than the optimized Markov binary cumulative sum chart that can detect a shift in p much faster than a Shewhart‐type chart. Copyright © 2014 John Wiley & Sons, Ltd.