Re-Evaluating the SPRT Chart with Estimated Process Parameters When the Underlying Distributions Are Gamma, Lognormal, and Weibull

Abstract

To reduce the impact of Phase-I parameter estimation on the performances of Phase-II control charts, researchers have incorporated the ideology of Guaranteed In-control Performance (GICP) in their statistical designs to limit the risk of excessive false alarms. At present, most research works have primarily focused on normally distributed data. However, the assumption of normality is often violated in manufacturing environments, and certain data may exhibit positively skewed distributions. In this paper, we investigate the performance of the SPRT control chart with estimated process parameters designed using the GICP method under three different skewed distributions, i.e., the Gamma, Lognormal, and Weibull distributions. The study is conducted by varying the Phase-I sample size and the degree of skewness in order to reveal their impacts upon the in-control and out-of-control performances of the SPRT chart with estimated process parameters. Results show that an increase in the skewness level leads to rapid deterioration in both the in-control and out-of-control expected values of the average time to signal (AATS) and the average standard deviation of the time to signal (ASDTS). Interestingly, we have found that increasing the Phase-I sample size leads to deterioration in the conditional in-control performance, but an improvement in the out-of-control AATS and ASDTS values. Furthermore, it is found that, among the three distributions, the Lognormal distribution produces the least stable performance when skewness is large and the Phase-I sample size is small.