The Bernoulli cumulative sum (CUSUM) chart has been shown to be effective for monitoring the rate of nonconforming items in high-quality processes where the in-control proportion of nonconforming items (p 0 ) is low. The implementation of the Bernoulli CUSUM chart is often based on the assumption that the in-control value p 0 is known; therefore, when p 0 is unknown, accurate estimation is necessary. We recommend using a Bayes estimator to estimate the value of p 0 to incorporate practitioner knowledge and to avoid estimation issues when no nonconforming items are observed in phase I. We also investigate the effects of parameter estimation in phase I on the upper-sided Bernoulli CUSUM chart by using the expected value of the average number of observations to signal (ANOS) and the standard deviation of the ANOS. It is found that the effects of parameter estimation on the Bernoulli CUSUM chart are more significant than those on the Shewhart-type geometric chart. The low p 0 values inherent to high-quality processes imply that a very large, and often unrealistic, sample size may be needed to accurately estimate p 0 . A methodology to identify a continuous variable to monitor is highly recommended when the value of p 0 is low and the required phase I sample size is impractically large.

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The problem of detecting changes in the parameter p in a Bernoulli process with two possible categories for each observation has been extensively investigated in the SPC literature, but there is much less work on detecting changes in the vector parameter p in a multinomial process where there are more than two categories. A few papers have considered the case in which the direction of the change in p is known, but there is almost no work for the important case in which the direction of the change is unknown and individual observations are obtained. This paper proposes a multinomial generalized likelihood ratio (MGLR) control chart based on a likelihood ratio statistic for monitoring p when individual observations are obtained and the direction and size of the change in p are unknown. A set of 2‐sided Bernoulli cumulative sum (CUSUM) charts is proposed as a reasonable competitor of the MGLR chart. It is shown that the MGLR chart has better overall performance than the set of 2‐sided Bernoulli CUSUM charts over a wide range of unknown shifts. Equations are presented for obtaining the control limit of the MGLR chart when there are three or four components in p.

Various generalized likelihood ratio (GLR) charts have been proposed to monitor count processes such as binomial, Bernoulli, Poisson, and multinomial processes. The advantages of GLR charts are that designing the chart is relatively easy, estimates of the process change‐point and shift size are available for post‐signal diagnosis, and they are effective in detecting a wide range of shifts in the process parameter. However, for some special cases of the observations, such as observing all defective items or all non‐defective items, the GLR chart statistic for monitoring a count process has been said to be undefined. We show that the GLR chart statistic is always well defined.

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