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.