Diameter and volume minimizing confidence sets in Bayes and classical problems. (English) Zbl 0741.62032
Summary: If \(X\sim P_ \theta\), \(\theta\in \Omega\), and \(\theta\sim G<<\mu\), where \(dG/d\mu\) belongs to the convex family
\[
\Gamma_{L,U}=\{g: L\leq cg\leq U,\hbox{ for some }c>0\},
\]
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 \(\mu<<\lambda<<\mu\).
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 \({\mathcal X}\) 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 \({\mathcal X}\), 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 C. Borell [Periodica Math. Hungar. 6, 111-136 (1975; Zbl 0402.28007)] is then used to conclude that diameter minimizing sets are spheres whenever the underlying distribution \(P\) is symmetric, absolutey continuous and the density \(f\) is such that \(f^{-1/m}\) is convex. All standard symmetric multivariate densities satisfy this conditions. Applications are made to several Bayes and classical problems and admissibility implications of these results are discussed.
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 \({\mathcal X}\) 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 \({\mathcal X}\), 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 C. Borell [Periodica Math. Hungar. 6, 111-136 (1975; Zbl 0402.28007)] is then used to conclude that diameter minimizing sets are spheres whenever the underlying distribution \(P\) is symmetric, absolutey continuous and the density \(f\) is such that \(f^{-1/m}\) is convex. All standard symmetric multivariate densities satisfy this conditions. Applications are made to several Bayes and classical problems and admissibility implications of these results are discussed.
MSC:
62F25 | Parametric tolerance and confidence regions |
62F15 | Bayesian inference |
62B99 | Sufficiency and information |
60D05 | Geometric probability and stochastic geometry |