Inefficient Best Invariant Tests

TL;DR

This paper studies the asymptotic power of invariant test statistics in high-dimensional models, showing they are often inefficient against certain alternatives, with applications to ANOVA and exponential families.

Contribution

It demonstrates that invariant tests can be asymptotically powerless against specific alternatives in high-dimensional settings, extending previous methods to new statistical models.

Findings

Invariant tests have asymptotic power equal to their level in high-dimensional models.

Application to ANOVA in the Neyman-Scott problem shows limitations of invariant tests.

Method based on Cibisov's approach is reviewed and extended.

Abstract

Test statistics which are invariant under various subgroups of the orthogonal group are shown to provide tests whose powers are asymptotically equal to their level against the usual type of contiguous alternative in models where the number of parameters is allowed to grow as the sample size increases. The result is applied to the usual analysis of variance test in the Neyman-Scott many means problem and to an analogous problem in exponential families. Proofs are based on a method used by Cibisov(1961) to study spacings statistics in a goodness-of-fit problem. We review the scope of the technique in this context.