Statistics Papers

Document Type

Journal Article

Date of this Version

1980

Publication Source

The Annals of Statistics

Volume

8

Issue

3

Start Page

572

Last Page

585

DOI

10.1214/aos/1176345009

Abstract

Two examples are presented. In each, p independent normal random variables having unit variance are observed. It is desired to estimate the unknown means, θi, and the loss is of the form L(θ,a) = (Σpi=1ν(θi))−1Σpi=1ν(θi)(θiai)2. The usual estimator, δ0(x)=x, is minimax with constant risk.

In the first example ν(t) = ert. It is shown that when r ≠ 0,δ0 is inadmissible if and only if p ⩾ 2 whereas when r = 0 it is known to be inadmissible if and only if p ⩾ 3.

In the second example ν(t)=(1+t2)r/2. It is shown that δ0 is inadmissible if p > (2−r)/(1−r) and admissible if p < (2−r)/(1−r). (In particular δ0 is admissible for all p when r ⩾ 1 and only for p = 1 when r < 0.) In the first example the first order qualitative description of the better estimator when δ0 is inadmissible depends on r, while in the second example it does not. An example which is closely related to the first example, and which has more significance in applications, has been described by J. Berger.

Keywords

estimation, admissibility, estimating several normal means

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Date Posted: 27 November 2017

This document has been peer reviewed.