LARGE-SAMPLE ASYMPTOTIC THEORY OF TESTS FOR UNIFORMITY ON THE GRASSMANN MANIFOLD

The Grassmann manifold G(k,m-k) consists of k-dimensional linear subsp aces V in R(m). To each V in G(k,m-k), corresponds a unique m x m orth ogonal projection matrix P idempotent of rank k. Let P-k,P-m-k denote the set of all such orthogonal projection matrices. We discuss distrib ution theory on P-k,P-m-k, presenting the differential form for the in variant measure and properties of the uniform distribution, and sugges t a general family F-(P) of non-uniform distributions. We are mainly c oncerned with large sample asymptotic theory of tests for uniformity o n P-k,P-m-k. We investigate the asymptotic distribution of the standar dized sample mean matrix U taken from the family F-(P) under a sequenc e of local alternatives for large sample size n. For tests of uniformi ty versus the matrix Langevin distribution which belongs to the family F-(P), we consider three optimal tests-the Rayleigh-style, the likeli hood ratio, and the locally best invariant tests. They are discussed i n relation to the statistic U, and are shown to be approximately, near uniformity, equivalent to one another. Zonal and invariant polynomial s in matrix arguments are utilized in derivations. (C) 1995 Academic P ress, Inc.