DIAMETER AND VOLUME MINIMIZING CONFIDENCE SETS IN BAYES AND CLASSICAL PROBLEMS

If X approximately P-theta, theta is-an-element-of OMEGA and theta approximately G << mu, where dG/d-mu belongs to the convex family GAMMA-L, U = {g: L less-than-or-equal-to cg less-than-or-equal-to U, for some c > 0}, then the sets minimizing lambda(S) subject to inf(G) is-an-element-of GAMMA-L, U P(G)(S/X) greater-than-or-equal-to p are derived, where P(G)(S/X) is the posterior probability of S under the prior G, and lambda is any nonnegative measure on OMEGA such that mu << lambda << mu. Applications are shown to several multiparameter problems and connectedness (or disconnectedness) of these sets is considered. The problem of minimizing the diameter is also considered in a general probabilistic framework. It is proved that if X is any finite-dimensional Banach space with a convex norm, and {P-alpha} is a tight family of probability measures on the Borel sigma-algebra of X, then there always exists a closed connected set minimizing the diameter under the restriction inf-alpha P-alpha(S) greater-than-or-equal-to p. It is also proved that if P is a spherical unimodal measure on R(m), then volume (Lebesgue measure) and diameter minimizing sets are the same. A result of Borell is then used to conclude that diameter minimizing sets are spheres whenever the underlying distribution P is symmetric absolutely continuous and the density f is such that f-1/m is convex. All standard symmetric multivariate densities satisfy this condition. Applications are made to several Bayes and classical problems and admissibility implications of these results are discussed.