Improved estimation for elliptically symmetric distributions with unknown block diagonal covariance matrix

Abstract

Let X, U₁, …, U_n-1 be n random vectors in ℝ^p with joint density of the form f((X - θ)^´∑^-1(X - θ) + ∑^n-1_{j = 1}U^´_j∑^-1U_j) where both θ∈ℝ^p and ∑ are unknown, the scale matrix ∑ being supposed structured as a diagonal matrix, that is, ∑= diag(∑₁, …,∑_b) where, for 1 ≤ i ≤ b, ∑_i is a p_i × p_i matrix and ∑_{i = 1}^bp_i = p. We consider the problem of the estimation of θ with the invariant loss (δ - θ)´∑^-1(δ - θ) and propose estimators which dominate the usual estimator δ₀(X) = X. These domination results hold simultaneously for the entire class of such distributions. The proof uses a generalization of integration by parts formulae by Stein and Haff. We also consider estimating ∑ under L_S(∑^,∑) = tr(∑^∑^-1) - log |∑^∑^-1| - p and propose estimators that dominate the unbiased estimator ∑^^UB = diag(S₁, …, S_b)/(n - 1), where S_i = ∑_{j = 1}^{n - 1}U_ijU´_ij and dim U_ji = p_i, for 1 ≤ i ≤ b and 1 ≤ j ≤ n - 1. The subsequent development of expressions is analogous to the unbiased estimators of risk technique and, in fact, reduces to an unbiased estimator of risk in the normal case.