Abstract
If $X \sim P_\theta, \theta \in \Omega$ and $\theta \sim G \ll \mu$, where $dG/d\mu$ belongs to the convex family $\Gamma_{L, U} = \{g: L \leq \operatorname{cg} \leq U$, for some $c > 0\}$, then the sets minimizing $\lambda(S)$ subject to $\inf_{G \in \Gamma_{L,U}} P_G(S\mid X) \geq p$ are derived, where $P_G(S\mid X)$ is the posterior probability of $S$ under the prior $G$, and $\lambda$ is any nonnegative measure on $\Omega$ such that $\mu \ll \lambda \ll \mu$. 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 $\mathscr{X}$ 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 $\mathscr{X}$, then there always exists a closed connected set minimizing the diameter under the restriction $\inf_\alpha P_\alpha(S) \geq p$. It is also proved that if $P$ is a spherical unimodal measure on $\mathbb{R}^m$, 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.
Citation
Anirban DasGupta. "Diameter and Volume Minimizing Confidence Sets in Bayes and Classical Problems." Ann. Statist. 19 (3) 1225 - 1243, September, 1991. https://doi.org/10.1214/aos/1176348246
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