Robust shrinkage estimation for elliptically symmetric distributions with unknown covariance matrix

Let X,V1,…,Vn−1 be n random vectors in Rp with joint density of the formf(X−θ)′Σ−1(X−θ)+∑j=1n−1 Vj′Σ−1Vj,where both θ and Σ are unknown. We consider the problem of the estimation of θ with the invariant loss (δ−θ)′Σ−1(δ−θ) and propose estimators which dominate the usual estimator δ0(X)=X simultaneously for the entire class of such distributions. The proof involves the development of expressions which are analogous to unbiased estimators of risk and which in fact reduce to unbiased estimators of risk in the normal case. The method is applicable to the case where Σ is structured. As an example, we examine the case where Σ is diagonal.