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May, 1980 Examples of Berger's Phenomenon in the Estimation of Independent Normal Means
L. Brown
Ann. Statist. 8(3): 572-585 (May, 1980). DOI: 10.1214/aos/1176345009


Two examples are presented. In each, $p$ independent normal random variables having unit variance are observed. It is desired to estimate the unknown means, $\theta_i$, and the loss is of the form $L(\theta, a) = (\Sigma^p_{i=1} \nu(\theta_i))^{-1} \Sigma^p_{i=1}\nu(\theta_i)(\theta_i - a_i)^2$. The usual estimator, $\delta_0(x) = x$, is minimax with constant risk. In the first example $\nu(t) = e^{rt}$. It is shown that when $r \neq 0, \delta_0$ is inadmissible if and only if $p \geqslant 2$ whereas when $r = 0$ it is known to be inadmissible if and only if $p \geqslant 3$. In the second example $\nu(t) = (1 + t^2)^{r/2}$. It is shown that $\delta_0$ is inadmissible if $p > (2 - r)/(1 - r)$ and admissible if $p < (2 - r)/(1 - r)$. (In particular $\delta_0$ is admissible for all $p$ when $r \geqslant 1$ and only for $p = 1$ when $r < 0.$) In the first example the first order qualitative description of the better estimator when $\delta_0$ 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.


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L. Brown. "Examples of Berger's Phenomenon in the Estimation of Independent Normal Means." Ann. Statist. 8 (3) 572 - 585, May, 1980.


Published: May, 1980
First available in Project Euclid: 12 April 2007

zbMATH: 0447.62009
MathSciNet: MR568721
Digital Object Identifier: 10.1214/aos/1176345009

Primary: 62C15
Secondary: 62C10, 62F10

Rights: Copyright © 1980 Institute of Mathematical Statistics


Vol.8 • No. 3 • May, 1980
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