## The Annals of Statistics

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

#### 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