Hornik Kurt, Grün Bettina
Institute for Statistics and Mathematics, WU Wirtschaftsuniversität Wien, Welthandelsplatz 1, 1020 Vienna, Austria.
Department of Applied Statistics, Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria.
J Multivar Anal. 2014 Apr;126(100):14-24. doi: 10.1016/j.jmva.2014.01.003.
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 [Formula: see text]-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 [Formula: see text], we show that the proper conjugate priors can be characterized by the behavior of the moment generating function of [Formula: see text] at the boundary of the natural parameter space, or the second-order tail behavior of [Formula: see text]. 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.
迪亚科尼斯和伊尔维萨克（1979年）给出了自然指数族分布的共轭先验为恰当的以及具有该族均值参数的线性后验期望性质的必要条件。如果自然参数空间等于所有[公式：见正文]维实数的集合，那么他们关于恰当性和线性后验期望的条件也是充分的。在本文中，他们的结果被扩展到刻画当自然参数空间有界时共轭先验何时是恰当的。对于自然指数族通过球面概率分布[公式：见正文]的特殊情况，我们表明恰当的共轭先验可以由[公式：见正文]的矩生成函数在自然参数空间边界处的行为，或者[公式：见正文]的二阶尾部行为来刻画。此外，我们表明如果这些族是非正则的，那么线性后验期望永远不成立。这个特殊情况的结果也被扩展到通过椭圆概率分布的自然指数族。
