Testing for Redundancy in Normal Mixture Analysis

Abstract

In this paper we deal with the problem of testing for redundancy of variables in a mixture of two p-variate normal populations. If the investigator is only interested in estimation of the parameters of the first q variables, the last p − q variables are said to be redundant when they can be safely discarded, i.e., when they carry no additional information about the parameters of interest. The main contribution of this work consists in finding the likelihood ratio test for this hypothesis. In general, the distribution of the test is nonstandard, so that bootstrap methods are employed to analyze its properties; as an illustration, the techniques are applied to a real-data problem in the field of quality control. However in a specific case, namely when the Mahalanobis distance is large, the usual χ² asymptotic approximation is appropriate; thus we study, by means of a simulation experiment, the relationship between the Mahalanobis distance and the limiting distribution of the test.

Key Words:

• Likelihood ratio test
• Mahalanobis distance
• Mixture of regressions
• Bootstrap

Mathematics Subject Classification:

• 62F03
• 62F05

Acknowledgment

I would like to thank an anonymous referee for valuable comments which helped to improve an earlier version of the paper.