Evaluation of Predictive Distributions based on Information and Entropy

本研究的目的是从信息论的观点探讨统计模型的优度评估问题。以下是1996年通过这一研究项目取得的主要成果。当特定的概率分布族不包含产生数据的分布时，提出了用于评估由各种估计程序构造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法，强大的，惩罚似然，海湾程序等方面的计算机密集型的方法的理论和实践方面的使用的引导模型评估问题进行了研究。2.探讨了临床试验中配对设计二元数据的等效性检验。共研究了8个检验，包括由GSK方法构建的检验、使用估计零方差的检验等。3.研究了一种求解二阶非线性椭圆型方程的数值封闭方法，该方法保证了L^* 误差界。利用高阶C^0-有限元近似解的后验误差估计，证明了我们可以得到高精度的保证L^* 误差界。4.将验证过程推广到与MHD平衡点相关的不可微非线性椭圆问题，构造了一个自动封闭解且保证误差界的计算算法. 5.研究了秩检验类的统一构造过程，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.