Likelihood Ratio Tests for a Change in the Multivariate Normal Mean

Abstract

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 Chernoff (1973) by the “faces” method and to find changes in the frequencies of pronouns in the plays of Shakespeare.

Key Words:

• Change point
• Binary segmentation
• Bonferroni inequality
• Extramultinomial variation