## 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,

**, and the loss is of the form**

*θ*_{i}**. The usual estimator,**

*L*(*θ*,*a*) = (Σ^{p}_{i}_{=1}*ν*(*θ*_{i}))^{−1}Σ^{p}_{i}_{=1}*ν*(*θ*_{i})(*θ*_{i}−*a*_{i})^{2}**, is minimax with constant risk.**

*δ*_{0}(*x*)=*x*In the first example** ν(t) = e^{rt}**. It is shown that when

**is inadmissible if and only if**

*r*≠ 0,*δ*_{0}**whereas when**

*p*⩾ 2**it is known to be inadmissible if and only if**

*r*= 0**.**

*p*⩾ 3In the second example ** ν(t)=(1+t^{2})^{r/2}**. It is shown that

**is inadmissible if**

*δ*_{0}**and admissible if**

*p*> (2−*r*)/(1−*r*)**. (In particular**

*p*< (2−*r*)/(1−*r*)**is admissible for all**

*δ*_{0}**when**

*p***and only for**

*r*⩾ 1**when**

*p*= 1**.) In the first example the first order qualitative description of the better estimator when**

*r*< 0**is inadmissible depends on**

*δ*_{0}**, 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.**

*r*#### Keywords

estimation, admissibility, estimating several normal means

#### Recommended Citation

Brown, L. D.
(1980).
Examples of Berger's Phenomenon in the Estimation of Independent Normal Means.
*The Annals of Statistics,*
*8*
(3),
572-585.
http://dx.doi.org/10.1214/aos/1176345009

**Date Posted:** 27 November 2017

This document has been peer reviewed.