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Optimized recentered confidence spheres for the multivariate normal mean. (English) Zbl 1362.62058

Summary: G. Casella and J. T. Hwang [J. Am. Stat. Assoc. 78, 688–698 (1983; Zbl 0562.62031)] introduced a broad class of recentered confidence spheres for the mean \(\boldsymbol{\theta}\) of a multivariate normal distribution with covariance matrix \(\sigma^2\mathbf{I}\), 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 \(\boldsymbol{\theta}=\mathbf{0}\). 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 \(\boldsymbol{\theta}=\mathbf{0}\), 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.

MSC:

62F25 Parametric tolerance and confidence regions
62C12 Empirical decision procedures; empirical Bayes procedures
62H12 Estimation in multivariate analysis

Citations:

Zbl 0562.62031
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