多元线性模型中的最优预测

【作者】 黄介武；

【导师】 喻胜华；

【作者基本信息】 中南大学 ， 概率论与数理统计， 2006， 硕士

【摘要】 多元线性模型是一类重要的统计模型，现已成为自然科学和社会科学各领域中不可缺少的数据分析工具．由于其应用广泛，对多元线性模型的研究已经变得十分重要也取得了一系列的研究成果．在对多元线性模型的研究中，一个重要的方面是研究多元线性模型中的最优预测问题．已有许多文献对这方面的问题进行了研究，本文研究探讨了任意秩多元线性模型中的最优线性预测量与最优齐线性预测量，带线性约束条件的任意秩多元线性模型与带线性约束条件的任意秩生长曲线模型中的条件最优线性无偏预测，多元线性模型中岭型预测与最优线性无偏预测的最优性比较问题，基于岭估计的最优预测量与经典预测量关于离差阵的最优性判别问题，任意秩生长曲线模型中最优线性无偏预测的稳健性问题，完善了多元线性模型中的最优预测的相关理论．全文共由五章构成：第一章简要叙述了多元线性模型的基本概念以及多元线性模型中最优预测的研究现状及研究成果，以及本文问题的产生与本文的主要工作．另外还简要介绍了对本文问题进行研究所必须的预备知识．第二章研究了任意秩多元线性模型中的最优线性预测与最优齐线性预测，并考虑了带线性约束的任意秩多元线性模型的条件最优线性无偏预测．第三章在一般多元线性模型中针对常见的有偏预测——岭型预测与最优线性无偏预测的最优性判别进行了讨论，并就基于岭估计的最优预测量与经典预测量关于离差阵的最优性判别进行了探讨．第四章研究了一般生长曲线模型中的最优线性预测，最优线性无偏预测，最优齐线性预测，并考虑了任意秩的一般生长曲线模型中在线性约束条件HB=0下的条件最优线性无偏预测．第五章对任意秩生长曲线模型的最优线性无偏预测的稳健性进行了探讨．更多还原

【Abstract】 Multivariate linear model is a branch of important statistical model. Now it has become an indispensable tool for data analysis in every field of natural science and social science. Because of its comprehensive application in many fields, the study of the multivariate linear model is becoming more important and many researchers have obtained some excellent achievements. In this paper we study the optimal linear predication and the optimal uniform linear predication in the multivariate linear model with arbitrary rank, we consider the conditional optimal linear unbiased predication in the multivariate linear model with arbitrary rank and the growth curve model with arbitrary rank.We investigate the optimality distinguish of the shrinked predication and the optimal linear unbiased predication in the multivariate linear model.Robustness of the optimal linear unbiased predication in the growth curve model are investigated .This thesis consists of five parts as following:In the first chapter,we introduces some basic concepts and theorems on multivariate linear model and the research status ,and the origin of the problem we discussed.And the main works of this paper are also simply introduced.In the second chapter,we study the multivariate linear model with arbitrary rank.We obtain the optimal linear predication, the optimal uniform linear predication and the conditional optimal linear unbiased predication.In the third chapter, we investigate the optimality distinguish of the shrinked predication and the optimal linear unbiased predication in the multivariate linear model.In the fourth chapter,we study the the growth curve model. the optimal linear predication, the optimal linear unbiased predication and the optimal linear unbiased predication are obtained .we investigate the conditional optimal linear unbiased predication in the growth curve model with arbitrary rank.In the fifth chapter robustness of the optimal linear unbiased predication in the growth curve model are investigated. Necessary and sufficient conditions for the predictor of linear predictable variable to be robust with respect to covariance matrices are obtained.更多还原