Annals of Statistics

General Admissibility and Inadmissibility Results for Estimation in a Control Problem

James O. Berger, L. Mark Berliner, and Asad Zaman

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Abstract

Let $\mathbf{X} = (X_1, \cdots, X_p)^t$ be an observation from a $p$-variate normal distribution with unknown mean $\mathbf{\theta} = (\theta_1, \cdots, \theta_p)^t$ and identity covariance matrix. We consider a control problem which, in canonical form, is the problem of estimating $\mathbf{\theta}$ under the loss $L(\mathbf{\theta, \delta}) = (\mathbf{\theta}^t \mathbf{\delta} - 1)^2$, where $\mathbf{\delta(x)} = (\delta_1(\mathbf{x}), \cdots, \delta_p(\mathbf{x}))^t$ is the estimate of $\mathbf{\theta}$ for a given $\mathbf{x}$. General theorems are given for establishing admissibility or inadmissibility of estimators in this problem. As an application, it is shown that estimators of the form $\mathbf{\delta(x)} = (|\mathbf{x}|^2 + c)^{-1}\mathbf{x} + |\mathbf{x}|^{-4}w(|\mathbf{x}|)\mathbf{x}$, where $w(|\mathbf{x}|)$ tends to zero as $|\mathbf{x}| \rightarrow \infty$, are inadmissible if $c > 5 - p$, but are admissible if $c \leq 5 - p$ and $\mathbf{\delta}$ is generalized Bayes for an appropriate prior measure. Also, an approximation to generalized Bayes estimators for large $|\mathbf{x}|$ is developed.

Article information

Source
Ann. Statist., Volume 10, Number 3 (1982), 838-856.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345875

Digital Object Identifier
doi:10.1214/aos/1176345875

Mathematical Reviews number (MathSciNet)
MR663436

Zentralblatt MATH identifier
0496.62005

JSTOR
links.jstor.org

Subjects
Primary: 62C15: Admissibility
Secondary: 62F10: Point estimation 62H99: None of the above, but in this section

Keywords
Admissibility inadmissibility control problem spherically symmetric estimators risk function generalized Bayes estimators

Citation

Berger, James O.; Berliner, L. Mark; Zaman, Asad. General Admissibility and Inadmissibility Results for Estimation in a Control Problem. Ann. Statist. 10 (1982), no. 3, 838--856. doi:10.1214/aos/1176345875. https://projecteuclid.org/euclid.aos/1176345875


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