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Likelihood ratio tests for a change in the multivariate normal mean. (English) Zbl 0589.62037

Summary: A sequence of independent multivariate normal vectors with equal but possibly unknown variance matrices are hypothesized to have equal mean vectors, and we wish to test that the mean vectors have changed after an unknown point in the sequence. The likelihood ratio test is based on the maximum Hotelling ${T}^{2}$ for the sequences before and after the change point.

The main result is a conservative approximation for its null distribution based on an improved Bonferroni inequality. If the change is judged significant, then further changes are estimated by splitting the two subsequences formed by the first change point.

The methods can also be used to test for a change in row probabilities of a contingency table, allowing for extramultinomial variation. The results are used to find changes in a set of geological data previously analyzed by H. Chernoff [The use of faces to represent points in k- dimensional space graphically. ibid. 68, 361-368 (1973)] by the ”faces” method and to find changes in the frequencies of pronouns in the plays of Shakespeare.

##### MSC:
 62H15 Multivariate hypothesis testing 62E20 Asymptotic distribution theory in statistics