  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.
