The Moments of the Optimal Average Run Length of the Multivariate Exponentially Weighted Moving Average Control Chart For Equally Correlated Variables

Abstract

The Hotelling’s T² is a well-known statistic for detecting a shift in the mean vector of a multivariate normal distribution. Control charts based on T² have been used in statistical process control for monitoring a multivariate process. Although it is a powerful tool, the T² statistic is deficient when the shift to be detected in the mean vector of a multivariate process is small and consistent. The Multivariate Exponentially Weighted Moving Average (MEWMA) control chart is one of the control statistics used to overcome the drawback of the Hotelling’s T² statistic. In this paper, the distribution of the Average Run Length (ARL) of the (MEWMA) control chart when the quality characteristics exhibit substantial cross correlation and when the process is in control and out-of-control was derived using the Markov Chain algorithm. The derivation of the probability functions and the moments of the run length distributions were also obtained and they were consistent with some existing results for the in – control and out –of –control situation. By simulation process, the procedure identified a class of ARL for the MEWMA control chart when the process is in –control and out- of – control. From our study it was observed that the MEWMA scheme is quite adequate for detecting a small shift and a good way to improve the quality of goods and services in a multivariate situation. It was also observed that as the in-control average run length ARL₀ and the number of variables ( p) increases, the optimum value of the ARL_opt increases asymptotically and as the magnitude of the shift  increases, the optimal ARL_opt decreases. Finally we use examples from the literature to illustrate our method and demonstrate its efficiency.

Keywords: Moments, Average Run Length, multivariate exponentially weighted moving average, Markov Chain, optimal smoothing parameter.