Multivariate Symmetry Models

Abstract

Let Γ0 be a fixed, compact subgroup of the group Γ of orthogonal transformations on R d . A random variable x, with values in R d and distribution P, is Γ 0 -symmetric if x and γx have the same distribution for all γ ∈Γ0. In terms of P, this means P(A) = P(ΓA) for all Borel sets A and all γ ∈Γ0. Let x1,…,x n be iid random variables with values in R d . The Γ0-symmetry model asserts that the x1 have an unknown common distribution that is Γ0-symmetric. The Γ 0 -location model specifies that for some unknown η∈R d , the random variables x 1-η,…, x n-η have an unknown common distribution P which is Γ0-symmetric. This paper develops some methods of inference for these multivariate symmetry models. Unlike the one dimensional case, there are a large number of “symmetry” notions in R d ,d > 1; Section 2.3 provides a few simple, useful examples, which figure in subsequent development.