Design and Implementation of Two CUSUM Schemes for Simultaneously Monitoring the Process Mean and Variance with Unknown Parameters

Designing joint monitoring schemes for the mean and variance of a Gaussian process (normal distribution) using a single combined statistic instead of the traditional approach of using two separate statistics has gained many attention in recent years. Most of the existing one-chart schemes, however, assume that the true process parameters (standards) are known which is usually not practical and will lead to problems because of improper choices of statistic and control limits. In this paper, we propose two CUSUM control schemes that instinctively work well for the joint monitoring in the case of the unknown parameters by correcting the influence of the reference sample on the plotting statistic. We provide the control limits of the proposed control charts for practical implementation and also offer follow-up procedures for post-signal detection of the nature of shifts. We carry out a comprehensive simulation study to examine the performance of the schemes. When the true population parameters are unknown, we observe clear and distinct performance advantages of the proposed schemes. The empirical design issues regarding the optimal choice of the reference value of the proposed CUSUM schemes are systematically investigated. We provide beneficial recommendations for the practitioners. We also provide an example to illustrate the practical relevance of the proposed schemes. Copyright (C) 2016 John Wiley & Sons, Ltd.