A new recentered confidence sphere for the multivariate normal mean

TL;DR

This paper introduces a new recentered confidence sphere for the multivariate normal mean, optimizing the radius function to improve coverage probability and expected volume compared to existing methods.

Contribution

It proposes a novel radius function for the confidence sphere, minimizing expected volume at zero mean while maintaining coverage, advancing confidence set construction.

Findings

Improved minimum coverage probability over previous methods.

Reduced scaled expected volume at theta=0.

Comparable or better performance in coverage and volume metrics.

Abstract

We describe a new recentered confidence sphere for the mean, theta, of a multivariate normal distribution. This sphere is centred on the positive-part James-Stein estimator, with radius that is a piecewise cubic Hermite interpolating polynomial function of the norm of the data vector. This radius function is determined by numerically minimizing the scaled expected volume, at theta = 0, of this confidence sphere, subject to the coverage constraint. We use the computationally-convenient formula, derived by Casella and Hwang [3], for the coverage probability of a recentered confidence sphere. Casella and Hwang, op. cit., describe a recentered confidence sphere that is also centred on the positive-part James-Stein estimator, but with radius function determined by empirical Bayes considerations. Our new recentered confidence sphere compares favourably with this confidence sphere, in terms of both the minimum coverage probability and the scaled expected volume at theta = 0.