A weak functional framework for applications in statistics

Abstract

For the problem of estimating a general loss of the form c(‖x−θ‖²), Stein’s identity is particularly relevant in deriving unbiased estimators of loss when x is used as an estimate of θ and is distributed as $\mathcal{N}_{p}(\theta,I)$, and when c is the identity function. In [3], Fourdrinier and Lepelletier show that extensions to other distributions (actually, to spherically symmetric distributions) and to general functions c are conceivable, but through another approach involving a Green’s formula. Somewhat surprisingly, the statistical context induces an unusual weak functional framework. The main goal of this paper is to present such an analytic context.