Multivariate statistical process control charts based on the approximate sequential χ² test

Abstract

Similar to the univariate CUSUM chart, a multivariate CUSUM (MCUSUM) chart can be designed to detect a particular size of the mean shift optimally based on the scheme of a sequential likelihood ratio test for the noncentrality parameter. However, in multivariate case, the probability ratio of a sequential test is intractable mathematically and the test statistic based on the ratio does not have a closed form expression which makes it impractical for real application. We drive an approximate log-likelihood ratio and propose a multivariate statistical process control chart based on a sequential χ² test to detect a change in the noncentrality parameter. The statistical properties of the proposed test statistic are explored. The average runs length (ARL) performance of the proposed charts is compared with other MCUSUM charts for process mean monitoring. The experimental results reveal that the proposed charts achieve superior, both zero-state and steady-state, ARL performance over a wide range of mean shifts, especially when the dimension of measurements is large.

Keywords:

• likelihood ratio test
• multivariate CUSUM
• sequential χ² test
• statistical process control

Acknowledgements

The authors are very grateful to the editor and two anonymous referees for their valuable and constructive comments and suggestions which greatly improved the final version of the paper.

Funding

This work was supported by Qatar National Research Fund (QNRF) [grant number NPRP 08-323-2-116].

Conflict of interest

The statements made herein are solely the responsibility of the authors.