On standard conjugate families for natural exponential families with bounded natural parameter space

JOURNAL OF MULTIVARIATE ANALYSIS

卷:126

Article

Hornik, Kurt¹  Gruen, Bettina² 

[1] WU Wirtschaftsuniv Wien, Inst Stat & Math, A-1020 Vienna, Austria

[2] Johannes Kepler Univ Linz, Dept Appl Stat, A-4040 Linz, Austria

关键词: Bayesian analysis;    Conjugate prior;    Elliptical distribution;    Exponential family;    Linear posterior expectation;    Spherical distribution;   

DOI  :  10.1016/j.jmva.2014.01.003

来源: Elsevier

【 摘 要 】

Diaconis and Ylvisaker (1979) give necessary conditions for conjugate priors for distributions from the natural exponential family to be proper as well as to have the property of linear posterior expectation of the mean parameter of the family. Their conditions for propriety and linear posterior expectation are also sufficient if the natural parameter space is equal to the set of all d-dimensional real numbers. In this paper their results are extended to characterize when conjugate priors are proper if the natural parameter space is bounded. For the special case where the natural exponential family is through a spherical probability distribution eta, we show that the proper conjugate priors can be characterized by the behavior of the moment generating function of eta at the boundary of the natural parameter space, or the second-order tail behavior of eta. In addition, we show that if these families are non-regular, then linear posterior expectation never holds. The results for this special case are also extended to natural exponential families through elliptical probability distributions. (C) 2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).

【 授权许可】

Free