Fiducial inference in the pivotal family of distributions  

作　　者：XU Xingzhong & LI Guoying Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China 

出　　处：《Science China Mathematics》2006年第3期410-432,共23页中国科学：数学（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant Nos.10271013,10071090).

摘　　要：In this paper a family, called the pivotal family, of distributions is considered.A pivotal family is determined by a generalized pivotal model. Analytical results show that a great many parametric families of distributions are pivotal. In a pivotal family of distributions a general method of deriving fiducial distributions of parameters is proposed. In the method a fiducial model plays an important role. A fiducial model is a function of a random variable with a known distribution, called the pivotal random element, when the observation of a statistic is given.The method of this paper includes some other methods of deriving fiducial distributions. Specially the first fiducial distribution given by Fisher can be derived by the method. For the monotone likelihood ratio family of distributions, which is a pivotal family, the fiducial distributions have a frequentist property in the Neyman-Pearson view. Fiducial distributions of regular parametric functions also have the above frequentist property. Some advantages of the fiducial inference are exhibited in four applications of the fiducial distribution. Many examples are given, in which the fiducial distributions cannot be derived by the existing methods.In this paper a family, called the pivotal family, of distributions is considered. A pivotal family is determined by a generalized pivotal model. Analytical results show that a great many parametric families of distributions are pivotal. In a pivotal family of distributions a general method of deriving fiducial distributions of parameters is proposed. In the method a fiducial model plays an important role. A fiducial model is a function of a random variable with a known distribution, called the pivotal random element, when the observation of a statistic is given. The method of this paper includes some other methods of deriving fiducial distributions. Specially the first fiducial distribution given by Fisher can be derived by the method. For the monotone likelihood ratio family of distributions, which is a pivotal family, the fiducial distributions have a frequentist property in the Neyman-Pearson view. Fiducial distributions of regular parametric functions also have the above frequentist property. Some advantages of the fiducial inference are exhibited in four applications of the fiducial distribution. Many examples are given, in which the fiducial distributions cannot be derived by the existing methods.

关 键 词：CONFIDENCE bounds  fiducial distributions  fiducial model  frequentist prop-erty  GENERALIZED pivotal model  pivotal FAMILY of distributions  testing hypotheses. 

分 类 号：O212[理学—概率论与数理统计]