Research on derivations of Bayes estimators with decision-theoretical optimality and their applications

决策理论最优性贝叶斯估计量的推导及其应用研究

• 批准号: 16500172
• 负责人: KUBOKAWA Tatsuya
• 金额: $ 2.41万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16500172/
• 关键词: Bayesian method; statistical decision theory; minimaxity; hierarchical Bayes model; linear mixed model; small area estimation; parametric restriction; variance component; 経験ベイズ推定; 情報量規準; 信頼区間; F検定; 高次元解析; 判別分析; 階層的ベイズ推定; 線形回帰モデル; 変数選択; 共分

The usefulness of the Bayesian procedures has been recently recognized from practical aspects. In this research project, I have shown the optimality of the Bayesian procedures from a decision-theoretic view point in several statistical problems as well as the usefulness in applications. The details are given below:1) In the estimation of a mean vector of a multivariate normal distribution, the characterization of the prior distributions has been given so that the resulting Bayes estimator is minimax and/or admissible. When the prior distribution has a hierarchical structure, I have derived conditions under which the hierarchical Bayes estimators are minimax.2) In the estimation of the component of covariance matrix in a multivariate linear mixed model, I have established a unified theory for the improvement through the truncated method. This problem is related to the estimation of the covariance matrices under the inequality restriction. I have considered several estimation problems under parametric restrictions and have shown the dominance results of Bayesian estimators. Also I have obtained the empirical Bayes estimator of the covariance matrix in the high dimensional cases and shown the theoretical optimality as well as the practical usefulness in data analysis.3) In the nested error regression model, I have derived the information criterion for selecting explanatory variables. This model is useful in the small area problem, and I have constructed an asymptotically corrected confidence interval of the small area mean and an asymptotically corrected test statistic for the linear hypothesis.

贝叶斯程序的实用性最近已从实践方面得到承认。在这个研究项目中，我已经从决策理论的角度在几个统计问题以及在应用中的有用性贝叶斯程序的最优性。1）在多元正态分布均值向量的估计中，给出了先验分布的特征，使得得到的Bayes估计是极小极大和/或可容许的。当先验分布具有分层结构时，本文给出了分层Bayes估计为极小极大的条件。2）在多元线性混合模型协方差阵分量的估计中，本文建立了用截尾法改进的统一理论。这个问题涉及到不等式约束下协方差矩阵的估计。我已经考虑了参数限制下的几个估计问题，并显示了贝叶斯估计的优势结果。在高维情形下，得到了协方差阵的经验Bayes估计，证明了理论上的最优性以及在数据分析中的实用性。3）在嵌套误差回归模型中，推导了选择解释变量的信息准则。该模型在小区域问题中是有用的，并且我构造了小区域均值的渐近校正置信区间和线性假设的渐近校正检验统计量。

项目成果

Estimation in a linear regression model under the Kullback-Leibler loss and its application to model selection

Kullback-Leibler 损失下线性回归模型的估计及其在模型选择中的应用

• 发表时间: 2007
• 期刊: J.Statistical Planning and Inference 137
• 作者: Mostafa Al Masum Shaikh;Helmut Prendinger;Mitsuru Ishizuka;和泉 諭;T.Kubokawa and H.Tsukuma
• 通讯作者: T.Kubokawa and H.Tsukuma

Linear Mixed Model and Small Area Estimation

线性混合模型和小面积估计

• 发表时间: 2007
• 作者: T.Kubokawa;M.S.Srivastava;T.Kubokawa;久保川 達也;T. Kubokawa
• 通讯作者: T. Kubokawa

経験ベイズ信頼区間の漸近補正と小地域推定への応用

经验贝叶斯置信区间的渐近修正及其在小区域估计中的应用

• 发表时间: 2005
• 期刊: 日本統計学会誌 シリーズJ 35-1
• 作者: 笹瀬吉隆;久保川達也
• 通讯作者: 久保川達也

Minimax multivariate empirical Bayes estimators under multicollinearlity

多重共线性下的极小极大多元经验贝叶斯估计

• 发表时间: 2005
• 期刊: Journal of Multivariate Analysis 93
• 作者: T.Kubokawa;M.-T.Tsai(共著);久保川 達也;M.S. Srivastava(共著);M. S. Srivastava and T. Kubokawa
• 通讯作者: M. S. Srivastava and T. Kubokawa

Estimation of covariance matrices in fixed and mixed effects linear models

固定效应和混合效应线性模型中协方差矩阵的估计

• 发表时间: 2006
• 期刊: Journal of Multivariate Analysis 97
• 作者: T.Kubokawa;M.-T.Tsai
• 通讯作者: M.-T.Tsai