On some shrinkage estimators of multivariate location. (English) Zbl 0564.62029

Sen, Pranab Kumar; Saleh, A. K. Md. Ehsanes

Ann. Stat. 13, 272-281 (1985).

Let \(F_{\theta}\) be a p (\(\geq 1)\) variate continuous distribution function (d.f.) defined on the Euclidean space \(E^ p\). The d.f. \(F_{\theta}\) is assumed to be diagonally symmetric about its location \(\theta =(\theta_ 1,...,\theta_ p)^ T\). In a shrinkage estimation problem, one has, a priori, some reason to believe that \(\theta\) is likely to lie in a small region containing a specified point \(\theta_ 0\) (which may be assumed to be zero without loss of generality).
Using a preliminary test for the hypothesis \(\theta =0\), a variant form of the James-Stein type estimation rule is used to formulate some shrinkage estimators of location based on rank statistics and U- statistics. A smaller critical value for the preliminary test has been recommended. In an asymptotic set up, the relative risks for these shrinkage estimators are shown to be smaller than their classical counterparts.

Reviewer: V.P.Gupta

MSC:

62G05	Nonparametric estimation
62C15	Admissibility in statistical decision theory
62G99	Nonparametric inference
62H12	Estimation in multivariate analysis

Keywords:

diagonally symmetric multivariate distribution; local alternatives; robustness; asymptotic risk; preliminary test; James-Stein type estimation rule; shrinkage estimators of location; rank statistics; U- statistics; critical value; relative risks