Estimating a general function of a quadratic function

Abstract

Let $x \in \mathbb{R}^{p}$ be an observation from a spherically symmetric distribution with unknown location parameter $\theta \in \mathbb{R}^{p}$. For a general non-negative function c, we consider the problem of estimating c(||x  −  θ||²) under the usual quadratic loss. For p  ≥  5, we give sufficient conditions for improving on the unbiased estimator γ₀ of c(||x  − θ||²) by competing estimators γ _s  =  γ₀  +  s correcting γ₀ with a suitable function s. The main condition relies on a partial differential inequality of the form k Δs  +  s ²  ≤  0 for a certain constant k  ≠  0. Our approach unifies, in particular, the two problems of quadratic loss estimation and confidence statement estimation and allows to derive new results for these two specific cases. Note that we formally establish our domination results (that is, with no recourse to simulation).