Large Sample Asymptotic Theory of Tests for Uniformity on the Grassmann Manifold

The Grassmann manifold Gk,m − k consists of k-dimensional linear subspaces V in Rm. To each V in Gk,m − k, corresponds a unique m × m orthogonal projection matrix P idempotent of rank k. Let Pk,m − k denote the set of all such orthogonal projection matrices. We discuss distribution theory on Pk,m − k, presenting the differential form for the invariant measure and properties of the uniform distribution, and suggest a general family F(P) of non-uniform distributions. We are mainly concerned with large sample asymptotic theory of tests for uniformity on Pk,m − k. We investigate the asymptotic distribution of the standardized sample mean matrix U taken from the family F(P) under a sequence of local alternatives for large sample size n. For tests of uniformity versus the matrix Langevin distribution which belongs to the family F(P), we consider three optimal tests-the Rayleigh-style, the likelihood ratio, and the locally best invariant tests. They are discussed in relation to the statistic U, and are shown to be approximately, near uniformity, equivalent to one another. Zonal and invariant polynomials in matrix arguments are utilized in derivations.