Confidence sets based on the positive part James–Stein estimator with the asymptotically constant coverage probability

Abstract

The asymptotic expansions for the coverage probability of a confidence set centred at the James–Stein estimator presented in our previous publications show that this probability depends on the non-centrality parameter τ² (the sum of the squares of the means of normal distributions). In this paper we establish how these expansions can be used for a construction of confidence region with constant confidence level, which is asymptotically (the same formula for both case τ→0 and τ→∞) equal to some fixed value 1−α. We establish the shrinkage rate for the confidence region according to the growth of the dimension p and also the value of τ for which we observe quick decreasing of the coverage probability to the nominal level 1−α. When p→∞ this value of τ increases as O(p^1/4). The accuracy of the results obtained is shown by the Monte-Carlo statistical simulations.

Keywords:

• confidence sets
• positive part James–Stein estimator
• multivariate normal distribution
• coverage probability
• asymptotical expansions
• second-order asymptotic

AMS Subject Classification:

• Primary: 62E20
• Secondary: 62F10

Acknowledgement

The authors would like to thank reviewers for useful remarks that strongly improved the presentation.