Invariant Tests for Means with Covariates

Abstract

We consider the problem of testing a hypothesis about the means of a subset of the components of a multivariate normal distribution with unknown covariance matrix, when the means of a second subset (the covariates) are known. Because of the possible correlation between the two subsets, information provided by the second subset can be useful for inferences about the means of the first subset. In this paper attention is restricted to the class of procedures invariant under the largest group of linear transformations which leaves the problem invariant. The family of tests which are admissible within this class is characterized. This family excludes several well-known tests, thereby proving them to be inadmissible (among all tests), while the admissibility (among invariant tests) of other tests is demonstrated. The powers of the likelihood ratio test LRT, the $D^2_{p+q} - D^2_p$ test, and the overall $T^2$ test are compared numerically; the LRT is deemed preferable on the basis of power and simplicity.