## The Annals of Statistics

- Ann. Statist.
- Volume 23, Number 4 (1995), 1408-1432.

### A New General Method for Constructing Confidence Sets in Arbitrary Dimensions: With Applications

A. DasGupta, J. K. Ghosh, and M. M. Zen

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 23, Number 4 (1995), 1408-1432.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324715

**Digital Object Identifier**

doi:10.1214/aos/1176324715

**Mathematical Reviews number (MathSciNet)**

MR1353512

**Zentralblatt MATH identifier**

0839.62028

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F25: Tolerance and confidence regions

Secondary: 60E15: Inequalities; stochastic orderings 62C10: Bayesian problems; characterization of Bayes procedures

**Keywords**

Confidence set star unimodal star-shaped sets Minkowski functional invariant sets level sets prior posterior HPD sets

#### Citation

DasGupta, A.; Ghosh, J. K.; Zen, M. M. A New General Method for Constructing Confidence Sets in Arbitrary Dimensions: With Applications. Ann. Statist. 23 (1995), no. 4, 1408--1432. doi:10.1214/aos/1176324715. https://projecteuclid.org/euclid.aos/1176324715