CUSUM Control Charts for Censored Gamma Distributed Quality Measurements

Abstract

Many companies conduct life tests on their products to ascertain their quality. In some cases, the tests are performed under accelerated circumstances and very often under censoring. The gamma distribution is widely used as a viable model for the lifetimes. In this thesis, we develop likelihood ratio cumulative sum (CUSUM) control charts for Type I right-censored data to monitor the mean lifetime of a product. The primary focus of this research centers on monitoring changes in the scale parameter η of the gamma distribution while the shape parameter β is kept fixed. A simulation study was conducted to assess the effect of sample size and censoring rate on the chart’s average run length (ARL) to detect changes in the scale parameter. We use two methods to calculate the ARL of the chart: a simulation method and a Markov chain method. In the first method, we follow Dickinson et al. (2014) by simulating data repeatedly until the chart reaches an out-of-control state then take the average, whereas in the second method we follow Lucas and Saccucci (1990) and Runger and Prabhu (1996) and use a Markov chain method along with the empirical cumulative distribution function (ECDF). Finally, we assess the off-target performance by finding the distribution and the probability of false alarms following Huh (2010)’s work. From the simulation method and the Markov chain method, we found that the CUSUM charts detect a change quickly if we increase the sample size, and increase or decrease the scale parameter. While from the probability of false alarms, the CUSUM charts perform well for small transition times t.