RESARCH ON THE THEORY AND APPLICATIONS OF EFFICIENT BAYES ESTIMATORS IN MULTIVARIATE STATISTICAL MODELS

多元统计模型中有效贝叶斯估计量的理论与应用研究

• 批准号: 11680320
• 负责人: KUBOKAWA Tatsuya
• 金额: $ 1.54万
• 依托单位: 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: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11680320/
• 关键词: multivariate regression models; multivariate normal distribution; shrinkage estimation; statistical decision theory; Stein problem; covariance matrix; empirical Bayes method; minimaxity; 線形回帰モデル; 変量効果; 多変量線線回帰モデル; 混合モデル; 変量モデル; 統計的推測; 推定

In this research project, I surveyed various estimation problems in multivariate statistical models from Theoretical and practical points of view, clarified decision-theoretic results such as admissibility and minimaxity and derived Bayes or shrinkage estimators more efficient than usual procedures.Especially, I made an exhaustive survey research paper covering estimation of mean vectors, mean matrices and covariance matrices of multivariate normal distributions and their extensions to non-normal distributions and estimation of ordered parameters and common parameters. This paper also gives new applicable examples of Stein type shrinkage procedures : one of them is to use empirical Bayes estimators in multicollinearity cases in linear regression models, which provide more efficient and stable estimates than the usual least squares method. The others include not only the derivation of a new variable selection procedure based on the shrinkage method, but also the improved estimators of t … More he noncentrality parameter and the multiple correlation coefficient and their applications to modifying the Mallows statistic and the usual adjusted R-square statistic.Some innovative theoretical results were obtained in this research project. For the estimation of a regression coefficients matrix in a multivariate linear regression model, I derived shrinkage estimators having smaller risks than the least squares estimator and showed the robustness of the improvement within the class of elliptically contoured distributions. This problem is interpreted as a prediction issue in a multivariate mixed linear model and it can be reduced to estimation of ratio of ordered covariance matrices of Wishart distributions. Using this idea and the arguments employed in estimating the covariance matrix, I derived several types of shrinkage estimators improving on the empirical Bayes or Efron-Morris estimator. For the estimation of the covariance matrix in the multivariate linear regression model, on the other hand, I succeeded in resolving a difficult problem, that is, I obtained two methods giving superior and minimax estimators which were constructed by using information contained in the estimator of the regression coefficients. Less

在这个研究项目中，我从理论和实践的角度调查了多元统计模型中的各种估计问题，澄清了可容许性和极小性等决策理论结果，并推导出比通常方法更有效的贝叶斯或收缩估计器。特别是对多元正态分布的均值向量、均值矩阵和协方差矩阵的估计及其在非正态分布中的推广、有序参数和公参数的估计进行了详尽的调查研究。本文还给出了Stein型收缩过程的新应用实例：其中之一是在线性回归模型的多重共线性情况下使用经验贝叶斯估计，它比通常的最小二乘法提供更有效和稳定的估计。本文不仅推导了一种新的基于收缩法的变量选择方法，而且改进了t的估计方法。此外，本文还研究了非中心性参数和多重相关系数及其在修正Mallows统计量和常用的调整r方统计量中的应用。本课题取得了一些具有创新性的理论成果。对于多元线性回归模型中回归系数矩阵的估计，我导出了比最小二乘估计具有更小风险的收缩估计，并在椭圆轮廓分布类中显示了改进的稳健性。这个问题可以解释为一个多元混合线性模型的预测问题，它可以简化为对Wishart分布的有序协方差矩阵的比值的估计。利用这一思想和估计协方差矩阵所采用的参数，我推导了几种类型的收缩估计器，改进了经验贝叶斯或Efron-Morris估计器。另一方面，对于多元线性回归模型中协方差矩阵的估计，我成功地解决了一个难题，即利用回归系数的估计量中包含的信息构造了两种给出上估计量和极大极小估计量的方法。少

项目成果

T.Kubokawa: "Estimation of variance and covariance components in elliptically contoured distributions"Journal of the Japan Statistical Society. 30,2. 199-232 (2000)

T.Kubokawa：“椭圆轮廓分布中方差和协方差分量的估计”日本统计学会杂志。

T.Kubokawa: "Estimation of variance and covariance components in elliptically contoured distributions."Journal of the Japan Statistical Society. Vol.30. 199-232 (2000)

T.Kubokawa：“椭圆轮廓分布中方差和协方差分量的估计。”日本统计学会杂志。

T.Kubokawa: "Shrinkage and modification techniques in estimation of variance and the related problems : A review."Communications in Statistics-Theory and Methods. Vol.28. 613-650 (1999)

T.Kubokawa：“方差估计中的收缩和修改技术及相关问题：综述。”统计理论与方法通讯。

T.Kubokawa,A.K.Md.E.Saleh and Y.Konno: "Bayes, minimax and nonnegative estimators of variance components under Kullback-Leibler loss"Journal of Statistical Planning and Inference. 86,1. 201-214 (2000)

T.Kubokawa、A.K.Md.E.Saleh 和 Y.Konno：“Kullback-Leibler 损失下方差分量的贝叶斯、极小极大和非负估计量”统计规划与推理杂志。

T.Kubokawa and M.S.Srivastava: "Robust improvement in estimation of a mean matrix in an elliptically contoured distribution"Journal of Multivariate Analysis. 76,1. 138-152 (2001)

T.Kubokawa 和 M.S.Srivastava：“椭圆轮廓分布中平均矩阵估计的稳健改进”多元分析杂志。