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August, 1995 A New General Method for Constructing Confidence Sets in Arbitrary Dimensions: With Applications
A. DasGupta, J. K. Ghosh, M. M. Zen
Ann. Statist. 23(4): 1408-1432 (August, 1995). DOI: 10.1214/aos/1176324715


Let $\mathbf{X}$ have a star unimodal distribution $P_0$ on $\mathbb{R}^p$. We describe a general method for constructing a star-shaped set $S$ with the property $P_0(\mathbf{X} \in S) \geq 1 - \alpha$, where $0 < \alpha < 1$ is fixed. This is done by using the Camp-Meidell inequality on the Minkowski functional of an arbitrary star-shaped set $S$ and then minimizing Lebesgue measure in order to obtain size-efficient sets. Conditions are obtained under which this method reproduces a level (high density) set. The general theory is then applied to two specific examples: set estimation of a multivariate normal mean using a multivariate $t$ prior and classical invariant estimation of a location vector $\mathbf{\theta}$ for a mixture model. In the Bayesian example, a number of shape properties of the posterior distribution are established in the process. These results are of independent interest as well. A computer code is available from the authors for automated application. The methods presented here permit construction of explicit confidence sets under very limited assumptions when the underlying distributions are calculationally too complex to obtain level sets.


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A. DasGupta. J. K. Ghosh. M. M. Zen. "A New General Method for Constructing Confidence Sets in Arbitrary Dimensions: With Applications." Ann. Statist. 23 (4) 1408 - 1432, August, 1995.


Published: August, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0839.62028
MathSciNet: MR1353512
Digital Object Identifier: 10.1214/aos/1176324715

Primary: 62F25
Secondary: 60E15 , 62C10

Keywords: confidence set , HPD sets , invariant sets , Level sets , Minkowski functional , Posterior , prior , star unimodal , star-shaped sets

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 4 • August, 1995
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