Consistency of the penalized MLE for two-parameter gamma mixture models

作     者：CHEN JiaHua LI ShaoTing TAN XianMing 

作者机构：Research Institute of Big Data Yunnan University Kunming 650221 China Department of Statistics University of British Columbia Vancouver VTC 5K5 Canada Department of Biostatistics University of North Carolina at Chapel Hill Chapel Hill NC 27514 USA 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2016年第59卷第12期

页      面：2301-2318页

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

基　　金：supported by Grants from One Thousand Talents at Yunnan University a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN–2014–03743) 

主　　题：constrained MLE identifiability finite mixture penalized likelihood Stirling formula 

摘      要：Two-parameter gamma distributions are widely used in liability theory, lifetime data analysis, financial statistics, and other areas. Finite mixtures of gamma distributions are their natural extensions, and they are particularly useful when the population is suspected of heterogeneity. These distributions are successfully employed in various applications, but many researchers falsely believe that the maximum likelihood estimator of the mixing distribution is consistent. Similarly to finite mixtures of normal distributions, the likelihood function under finite gamma mixtures is unbounded. Because of this, each observed value leads to a global maximum that is irrelevant to the true distribution. We apply a seemingly negligible penalty to the likelihood according to the shape parameters in the fitted model. We show that this penalty restores the consistency of the likelihoodbased estimator of the mixing distribution under finite gamma mixture models. We present simulation results to validate the consistency conclusion, and we give an example to illustrate the key points.