Quadratic loss estimation of a location parameter when a subset of its components is unknown

Abstract

We consider the problem of estimating the quadratic loss ||δ-θ||² of an estimator of the δ location parameter = (θ₁;...; θ_p) when a subset of the components of θ are restricted to be nonnegative. First, we assume that the random observation X is a Gaussian vector and, secondly, we suppose that the random observation has the form (X;U) and has a spherically symmetric distribution around a vector of the form (θ ; 0) with dim X = dim θ = p and dim U = dim 0 = k. For these two settings, we consider two location estimators, the least square estimator and a shrinkage estimators, and we compare theirs unbiased loss estimators with improved loss estimator.

Key words: Spherical symmetry; Quadratic loss; Least square estimator; Unbiased loss estimator; James-Stein estimation; Minimaxit