A NEW GENERAL-METHOD FOR CONSTRUCTING CONFIDENCE SETS IN ARBITRARY DIMENSIONS - WITH APPLICATIONS

Let X have a star unimodal distribution P-0 on R(p). We describe a gen eral method for constructing a star-shaped set S with the property P-0 (X is an element of S) greater than or equal to 1 - alpha, where 0 < a lpha < 1 is fixed. This is done by using the Camp-Meidell inequality o n the Minkowski functional of an arbitrary star-shaped set S and then minimizing Lebesgue measure in order to obtain size-efficient sets. Co nditions are obtained under which this method reproduces a level (high density) set. The general theory is then applied to two specific exam ples: set estimation of a multivariate normal mean using a multivariat e t prior and classical invariant estimation of a location vector thet a for a mixture model. In the Bayesian example, a number of shape prop erties of the posterior distribution are established in the process. T hese results are of independent interest as well. A computer code is a vailable from the authors for automated application. The methods prese nted here permit construction of explicit confidence sets under very l imited assumptions when the underlying distributions are calculational ly too complex to obtain level sets.