A New Chart for Monitoring the Covariance Matrix of Bivariate Processes

Abstract

In this article we consider a control chart based on the sample variances of two quality characteristics. The points plotted on the chart correspond to the maximum value of these two statistics. The main reason to consider the proposed chart instead of the generalized variance | S | chart is its better diagnostic feature, that is, with the new chart it is easier to relate an out-of-control signal to the variables whose parameters have moved away from their in-control values. We study the control chart efficiency considering different shifts in the covariance matrix. In this way, we obtain the average run length (ARL) that measures the effectiveness of a control chart in detecting process shifts. The proposed chart always detects process disturbances faster than the generalized variance | S | chart. The same is observed when the size of the samples is variable, except in a few cases in which the size of the samples switches between small size and very large size.

Keywords:

• Bivariate processes
• Control charts for monitoring the covariance matrix
• Generalized variance |S| chart
• Variable sample size

Mathematics Subject Classification:

• 1) Statistics and Computing
• 2) Probability Theory and Applications

Acknowledgments

This work was supported by CNPq—National Council for Scientific and Technological Development, Project 307744/2006-0—and FAPESP—The State of São Paulo Research Foundation, Project 2006/00491-0. We would like to thank the Associate Editor who handled the review process for this article, as well as the two anonymous referees who carefully read the earlier draft and made many constructive suggestions.

Notes

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*case I; **case II.

*case I; **case II.

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