## Electronic Journal of Statistics

### Optimized recentered confidence spheres for the multivariate normal mean

#### Abstract

Casella and Hwang, 1983, JASA, introduced a broad class of recentered confidence spheres for the mean $\boldsymbol{\theta}$ of a multivariate normal distribution with covariance matrix $\sigma^{2}\boldsymbol{I}$, for $\sigma^{2}$ known. Both the center and radius functions of these confidence spheres are flexible functions of the data. For the particular case of confidence spheres centered on the positive-part James-Stein estimator and with radius determined by empirical Bayes considerations, they show numerically that these confidence spheres have the desired minimum coverage probability $1-\alpha$ and dominate the usual confidence sphere in terms of scaled volume. We shift the focus from the scaled volume to the scaled expected volume of the recentered confidence sphere. Since both the coverage probability and the scaled expected volume are functions of the Euclidean norm of $\boldsymbol{\theta}$, it is feasible to optimize the performance of the recentered confidence sphere by numerically computing both the center and radius functions so as to optimize some clearly specified criterion. We suppose that we have uncertain prior information that $\boldsymbol{\theta}=\boldsymbol{0}$. This motivates us to determine the center and radius functions of the confidence sphere by numerical minimization of the scaled expected volume of the confidence sphere at $\boldsymbol{\theta}=\boldsymbol{0}$, subject to the constraints that (a) the coverage probability never falls below $1-\alpha$ and (b) the radius never exceeds the radius of the standard $1-\alpha$ confidence sphere. Our results show that, by focusing on this clearly specified criterion, significant gains in performance (in terms of this criterion) can be achieved. We also present analogous results for the much more difficult case that $\sigma^{2}$ is unknown.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 1798-1826.

Dates
First available in Project Euclid: 27 April 2017

https://projecteuclid.org/euclid.ejs/1493258584

Digital Object Identifier
doi:10.1214/17-EJS1272

Mathematical Reviews number (MathSciNet)
MR3641846

Zentralblatt MATH identifier
1362.62058

Subjects
Primary: 62F25: Tolerance and confidence regions
Secondary: 62J07: Ridge regression; shrinkage estimators

#### Citation

Abeysekera, Waruni; Kabaila, Paul. Optimized recentered confidence spheres for the multivariate normal mean. Electron. J. Statist. 11 (2017), no. 1, 1798--1826. doi:10.1214/17-EJS1272. https://projecteuclid.org/euclid.ejs/1493258584

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