A Generalized Likelihood Ratio Chart for Monitoring Bernoulli Processes

Abstract

This paper considers the problem of monitoring the proportion p of nonconforming items when a continuous stream of Bernoulli observations is available and the objective is to effectively detect a wide range of increases in p. The proposed control chart is based on a generalized likelihood ratio (GLR) statistic obtained from a moving window of past Bernoulli observations. The Phase II performance of this chart in detecting sustained increases in p is evaluated using the steady state average number of observations to signal. Comparisons of the Bernoulli GLR chart to the Shewhart CCC-r chart, the Bernoulli cumulative sum chart, and the Bernoulli exponentially weighted moving average chart show that the overall performance of the Bernoulli GLR chart is better than its competitors. In addition, methods are provided for designing the Bernoulli GLR chart so that this chart can be easily applied in practice.

A GLR control chart for monitoring a multinomial process

“…Here, it is sufficient to consider fewer shift directions than in Table 2, as all the components have the same in-control values. Table 3 gives the same conclusion about the set of two-sided Bernoulli CUSUM chart as we obtained from Table 2: The charts in column [4] with small predetermined shifts work well for shifts that are close to the tuning parameters, while the charts in column [6] work well for relatively large shifts.…”

“…The parameters under each column label are used for the set of two-sided Bernoulli CUSUM charts with same column labels in Table 2 and 3. The columns labeled [1] to [3] in Table 1 show the tuning parameters and control limits of the charts for the Columns [4] to [6] in Table 1 list the tuning parameters of the set of two-sided Bernoulli CUSUM charts that are designed to detect small, medium, and large shifts when p 0 = (.25, .25, .25, .25). As the in-control parameters are the same for all the four components, we use the same predetermined shifts in the 3 sets of two-sided Bernoulli CUSUM charts in each case of columns [4] to [6].…”

“…They showed that the Bernoulli CUSUM chart can detect shifts in p much faster than the Shewhart p ‐chart. Huang et al proposed a Bernoulli generalized likelihood ratio (GLR) chart to monitor p and showed that the Bernoulli GLR chart has better overall performance than the Bernoulli CUSUM chart over a wide range of possible shifts in p .…”

Latent change‐point detection in ordinal categorical data

“…Early work was reviewed by Xie et al and Topalidou and Psarakis . Recent developments for monitoring binomial data include Xie et al, Huang et al, Huang et al, and Wang and Reynolds . As to the monitoring of multinomial data, one can refer to the probability tree method with h − 1 stages for h categories developed by Duran and Albin and cumulative sum (CUSUM) chart based on likelihood ratio test suggested by Ryan et al To follow up, Weiß and Yashchin made more recent contributions to monitoring multinomial data.…”

GLR Charts for Simultaneously Monitoring a Sustained Shift and a Linear Drift in the Process Mean

“…GLR charts have some advantages that the size of the parameter change does not need to be specified and these charts have been shown to be very effective in a wide variety of settings in SPC applications. Recent investigations of this chart include Hawkins et al (2003), Runger and Testik (2003), Capizzi and Masarotto (2008), Zou et al (2009), Reynolds and Lou (2010), Huang et al (2012Huang et al ( , 2013, Xu et al (2012Xu et al ( , 2013, Wang and Reynolds (2013), and Reynolds et al (2013).…”