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Run-length distribution of multivariate control charts for covariance matrix with runs rules using finite Markov chain imbedding
Journal of the Korean Data & Information Science Society 2021;32:867-81
Published online July 31, 2021;
© 2021 Korean Data and Information Science Society.

Yuri Seo1 · Gyo-Young Cho2

12Department of Statistics, Kyungpook National University
Correspondence to: 1 Graduate student, Department of Statistics, Kyungpook National University, Daegu 41566, Korea
2 Professor, Department of Statistics, Kyungpook National University, Daegu 41566, Korea. E-mail:
Received June 2, 2021; Revised July 9, 2021; Accepted July 9, 2021.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hotelling's (1947) chi-square chart has a disadvantage in that it does not react sensitively to very small changes in the process, so we propose a multivariate chart with rules of various auxiliary runs to compensate for this. We use the finite Markov chain imbedding method and the rule of an auxiliary run to find the run-lengths distribution in a multivariate control chart. In this paper, we find the run-lengths distribution for the covariance matrix. At this time, it is assumed that the correlation coefficient according to the change of the covariance matrix does not change. We try to help evaluate the performance of the control chart by indicating not only the average run length but also the quartile.
Keywords : Average run length, Markov chain imbedding, multivariate control charts, run-length distribution.