Remarks on univariate elliptical distributions

Abstract

A random variable X is said to have an univariate elliptical distribution (or an elliptical density) with parameters μ and γ > 0 if it has a density of the form f_h(x|μ, γ) = γ^-1/2 h((x − μ)2/γ) for some function h(·). If μ = 0 and γ = 1, then X is said to have a spherical density corresponding to radial function h(·). Here we derive Chernoff-type inequality and a natural identity for univariate elliptical distribution. The problems of estimation of mean vector μ = (μ₁,…,μ_p) of a random vector X = (X₁,…,X_p) with independent components is discussed as an application of the identity. The risk of the improved estimator for p ≥ 3 dominating the unbiased estimator X of μ under squared error loss is derived. Locally asymptotic minimax estimation for a function g(θ) of θ = (μ, σ), σ² = γ is discussed. Special case of g(θ) = μ + cσ is discussed in detail and as a further special case, a locally asymptotic minimax estimator of μ + cσ is derived for normal distribution with parameters (μ, σ²). Finally Chernoff-type inequality and an identity are derived for a multivariate random vector X = (X₁,…,X_p) when the components X_i, 1 ≤ i ≤ p, are independent univariate elliptical random variables.