Inefficient Best Invariant Tests

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.