Phase II control chart for monitoring Gumbel’s bivariate exponential distribution

Abstract

Schemes for monitoring two or more time between events (TBE) data are becoming increasingly significant in statistical process monitoring. In practice, parameters must be estimated prior to the start of process monitoring. In this paper, we first propose a multivariate time between events (MTBE) control chart based on the one-sample log-likelihood ratio test for monitoring the Gumbel’s bivariate exponential (GBE) distribution, and then we analyze the effect of parameter estimation on this chart. To avoid the effect of parameter estimation, we then propose a method based on the two-sample log-likelihood ratio test for monitoring the shifts occurring in either the scale parameter or the dependence parameter or both for the GBE distribution process. The proposed method is designed with two-sample log-likelihood ratio test for univariate marginal distribution function and Gumbel copula function. A Max-type function-based approach is implemented for joint monitoring of margin distribution and copula-related structure. In contrast to the traditional schemes used to monitor bivariate processes, the strength of our proposed technique lies in the ability to identify exactly which component of the scale parameter or dependence parameter is responsible for the source of the signal. The performance of the proposed chart is analyzed via Monte Carlo simulations. Finally, two examples are mentioned to illustrate the implementation process of the proposed scheme.