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 $||\delta -\theta ||^{2}$ of an estimator $\delta $ of the location parameter $\theta =(\theta _{1},\ldots ,\theta _{p})$ when a subset of the components of $\theta$ 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 $(\theta ,0)$ with $\dim X=\dim \theta =p$ 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.