Properties of the T² Control Chart When Parameters Are Estimated

Abstract

Moments of the run length distribution are often used to design and study the performance of quality control charts. In this article the run length distribution of the T² chart for monitoring a multivariate process mean is analyzed. It is assumed that the in-control process observations are iid random samples from a multivariate normal distribution with unknown mean vector and covariance matrix. It is shown that the in-control run length distribution of the chart does not depend on the unknown process parameters. Furthermore, it is shown that the out-of-control run length distribution of the chart depends only on the statistical distance between the in-control and out-of-control mean vectors. It follows that a performance analysis can be given without knowledge of the in-control values of the parameters or their estimates. The performance of charts constructed using traditional F-distribution–based control limits is studied. Recommendations are given for sample size requirements necessary to achieve desired performance. Corrected control limits are given for designing charts with estimated parameters when large sample sizes are not available.

KEY WORDS :

• Average run length
• Control chart
• Hotelling's T²
• Run length