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

Let X, U1, …, Un-1 be n random vectors in ℝp with joint density of the form f((X - θ)´∑-1 (X - θ) + ∑n-1j = 1 U´j∑-1 Uj) where both θ∈ℝp 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 pi × pi matrix and ∑i = 1bpi = 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 LS(∑^,∑) = tr(∑^∑-1) - log |∑^∑-1| - p and propose estimators that dominate the unbiased estimator ∑^UB = diag(S1, …, Sb)/(n - 1), where Si = ∑j = 1n - 1Uij U´ij and dim Uji = pi, 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.