位置参数分布族中Fiducial分布与后验分布的关系

The Relationship Between Fiducial Distributions and Posterior Distributions in Location Family

本文考虑本质位置参数分布族中,参数的Fiducial分布与后验分布的等同问题．首先讨论了如何给出Fiducial分布,分析结果表明以分布函数形式给出Fiducial分布要比密度函数形式合理,同时,证明了所给的Fiducial分布具有频率性质．然后,研究在参数受到单侧限制时,Fiducial分布与后验分布等同的问题,给出的充要条件是分布族为指数分布族,此时,先验分布是一个广义先验分布,它不能被Lebesgue测度控制．最后,证明了在参数限制在一个有限区间内时,Fiducial分布与任何先验(包括广义先验分布)下的后验分布不等同．

In this paper, the family of distributions with an essential location parameter is considered. The focus of the paper is wether the fiducial distributions of location parameter are Bayes distributions, i.e., posterior distributions with some prior when the parametric space is not the whole real line. First how to represent fiducial distributions is discussed. The results are that fiducial distributions represented by cumulate distribution functions are reasonable, but by density functions are not, and the fiducial distributions given in this paper have a frequentist property. Then, that the fiducial distributions of location parameter truncted in one side to be the Bayes distributions is studied. A necessary and sufficient condition is that the population family of distributions is the family of exponential distributions, while the prior distribution is an improper prior which is not dominated by Lebesgue measure. Finally when the location parameter takes values in a finite interval, it is proved that fiducial distributions are not the Bayes distributions under any prior distribution including improper prior.