Evaluation of Predictive Distributions based on Information and Entropy

本研究的目的是从信息论的角度探讨统计模型的优劣评价问题。以下是该研究项目在1996年期间取得的主要成果。当指定的概率分布族不包含产生数据的分布时，提出了评估由各种估计程序构建的统计模型的信息准则。所提出的标准适用于评估模型估计的最大似然，稳健，惩罚似然，湾程序等。在计算机密集型方法方面，研究了在模型评估问题中使用自举法的理论和实践方面。2. 探讨了临床试验中采用配对设计的二元数据的等效检验。总共调查了8个检验，包括那些由GSK方法构建的检验，其中使用估计零方差的检验。3. 研究了一类二阶非线性椭圆型问题的保L^*误差边界数值封闭方法。利用后验误差估计方法对高阶C^0有限元问题的近似解进行了估计，得到了精度较高的保证L^*误差边界。4. 将验证过程推广到与MHD平衡有关的不可微非线性椭圆问题，构造了一种具有保证误差界的自动封闭解的计算算法。5. 研究了一类秩检验的统一构造过程，证明了由凸和距离导出的两样本问题的检验统计量的渐近正态性。

The purpose of the research is to investigate the problem of evaluating the goodness of statistical models from an information-theoretic point of view. The following are the main results obtained through this research project extending over 1996.1. Information criteria were proposed for evaluating statistical models constructed by various estimation procedures when the specified family of probability distributions does not contain the distribution generating the data. The proposed criteria are applied to the evaluation of models estimated by maximum likelihood, robust, penalized likelihood, Bays procedures, etc. In regard to computer-intensive methods the theoretical and practical aspects of the use of the bootstrap were investigated in model evaluation problems. 2. Equivalence tests for binary data with pair-matched design in clinical trials were explored. Eight tests were investigated altogether including those tests constructed by the GSK method, the tests which use estimated null variances among others. 3. A numerical enclosure method with guaranteed L^* error bounds for the solution of nonlinear elliptic problems of second order were investigated. Using an a posteriori error estimates for the approximate solution of the problem with higher order C^0-finite element, it was shown that we can obtain the guaranteed L^* error boundswith high accuracy. 4. The verification procedure were extended to nondifferentiable nonlinear elliptic problem related to MHD equilibra, and a computing algorithm which automatically encloses the solution with guaranteed error bounds was constructed. 5. Unified construction procedure of class of rank tests were investigated and the asymptotic normality was proved for test statistics for the two-sample problem derived by the convex sum distance.