To shrink or not to shrink: Hotelling's T~2 chart based on shrunken covariance estimates

摘要

Several authors have studied the effect of parameter estimation on the performance of Phase II control charts and shown that large in-control reference samples are necessary for the Phase II control charts to perform as desired. For higher dimensional data, even larger reference samples are required to achieve stable estimation of the in-control parameters. Shrinkage estimation has been widely studied as a method to achieve stable estimation of the covariance matrix for high-dimensional data. We investigate the average run length (ARL) distribution of the Hotelling T-2 chart when using a shrunken covariance matrix. Specifically, we explore the following questions: (1) Does the use of a shrinkage estimator of the covariance matrix result in reduced variability in the ARL performance of the T-2 chart? (2) Does the use of a shrinkage estimator of the covariance matrix result in a reduced occurrence of strictly multivariate false alarms on the T(2)chart? (3) How does shrinkage of the covariance matrix affect the out-of-control performance of the T-2 chart? We use a simulation study to investigate the use of shrinkage estimation with the Hotelling T-2 chart in Phase II. Our results indicate that, while shrinkage estimation affects the ARL performance of the T-2 chart, the benefits are small and occur in fairly specific circumstances. The benefits of shrinking may not justify the use of more advanced techniques.