Monitoring Markov Dependent Observations with a Log-Likelihood Based CUSUM

When control charts are used to monitor a proportion p it is traditionally assumed that the binary observations are independent. The work that has been done on monitoring autocorrelated binary observations has assumed a two-state Markov chain model with first-order dependence. We investigate the problem of monitoring p for such observations. We show that the most efficient chart for independent observations, the Bernoulli CUSUM chart, along with the traditional Shewhart chart, are not robust to autocorrelation. One approach to dealing with autocorrelation is to adjust the control limits of the traditional charts, but this does not produce the most efficient charts for detecting changes in p. We develop a more efficient log-likelihood-ratio based CUSUM chart for monitoring binary observations that follow the two-state Markov chain model. We show that this CUSUM chart can be well approximated by using a Markov chain that allows calculation of the properties of this chart. We also show that this CUSUM chart has better overall statistical performance than other charts available in the literature.