指数族分布中参数的经验贝叶斯估计

Empirical Bayes Estimators of Parameters in Exponential Family Distribution

指数族分布是一类应用广泛的分布类,包括了泊松分布、Gamma分布、Beta分布、二项分布等常见分布。在非寿险中,索赔额或索赔次数过程常常被假定服从指数族分布,由于风险的非齐次性,指数族分布中的参数θ也为随机变量,假定服从指数族共轭先验分布。此时风险参数的估计落入了Bayes框架,风险参数θ的Bayes估计被表达"信度"形式。然而,在实际运用中,由于先验分布与样本分布中仍然含有结构参数,根据样本的边际分布的似然函数估计结构参数,从而获得风险参数的经验Bayes估计,最后证明了该经验Bayes估计是渐近最优的。

Exponential family distribution is widely distributed class which include Poisson distribution, Gamma distribution, Beta distribution, Binomial distribution and the other common distribution. In non-life insurance, claims amount or claim frequency process are usually assumed to be exponential family distribution, because of the risk of homogeneity, the parameters of exponential family is assumed to be random variables, with exponential family conjugate prior distribution. Therefore, estimation of the risk parameters have fallen into the Bayes framework, and Bayes estimation can be expressed as samples of ＂credibility＂ forms. However, in the practical application, because some unknown structure parameters still exist in prior distribution and sample distribution. This paper suggest to use marginal distribution of the likelihood function to estimates structure parameters, and get corresponding empirical Bayes estimator. In addition, the empirical Bayes estimator is proved to be asymptotical optimal.