一、二阶矩和二阶中心矩同时估计的可容许性

鉴于回归系数月的最重要的估计量是观察值Y的线性函数,二阶中心矩尹和二阶原点矩护十尸的最重要的估计量是Y的非负定二次型,故限制在线性估计类中讨论月的可容许估计以及限制在非负定二次型估计类中讨论沪和尹十尸的可容许估计,越来越受到     

误差方差的非齐次估计的可容许的充要条件

研究误差方差的非齐次二次估计的可容许性．在平方损失下,给出了一个非齐次二次估计在非齐次二次估计类中是误差方差的容许估计的充要条件．     

协方差的非齐次估计的可容许性

本文研究协方差的非齐次二次估计的可容许性,在平方损失下,我们给出了一个非齐次二次估计在非齐次二次估计类中是协方差的容许估计的充要条件.     

线性模型中估计的可容许性

本文推广了La Motte有关线性模型中可容许估计的结果,由此解决了一种方式分组随机效应模型中方差分量二次及非负二次估计的可容许问题,并证明了可容许估计类是一最小完备类。     

具有P个方差分量的线性模型的Beyes二次估计及可容许估计的非负性

本文对具有 p 个方差分量的线性模型讨论了方差分量线性函数的 Bayes 不变二次估计问题,给出了 Bayes 不变二次估计(无偏和有偏)的显示表达式,并且证明了它们在各自考虑的类中形成了可容许估计的完全类.在可容许估计的完全类中,还讨论了非负参数函数的非负估计问题,给出了可容许的非负定估计存在的充要条件.     

正态线性模型中误差方差的二次型估计的容许性

设Ｙ遵从Ｎ（Ｘβ，σ２Ｉｎ），秩（Ｘ）＜ｎ，在平方损失下，本交给出σ２的二次型估计在整个估计类中可容许的充要条件．     

一、二阶矩同时估计的可容许性

考虑模型Y=(y_1,…,y_n)′=(β,…,β)′+(ε_1,…,ε_n)′=1β+ε.(1.1)此处1=(1,…,1)′;ε_1,…,ε_n 相互独立,E(ε_i)=0,E(ε_i~2)=σ~2,E(ε_i~3)=0,E(ε_i~4)=3σ~4,i=1,…,n;-∞<β<∞,0<σ<∞.鉴于 β 的最重要的估计量是观察值 Y 的线性函数,σ~2和 β~2+σ~2的最重要的估计量是 Y 的非负定二次型,在考虑 β 的估计时,首先把注意力集中在 Y 的线性函数上;在考虑σ~2或 β~2+σ~2的估计时,首先考虑 Y 的非负定二次型.参考文献[1]在一般线性模型和二次损失下,给出了回归系数的可估线性函数的估计在线性估计类中是可容许的充要条件.参考文献[2]和[3]在模型(1.1)和平方损失下给出了 σ~2的估计在非负定二次型估计类中是可容许的充要条件;而在一般线性模型和平方损失下,给出了 σ~2的估计在非负定二次型估计类中是可容许的必要条件和充分条件,给出了相当大的一类可容许估计;此外,给     

回归系数的非齐次线性估计的可容许性

考虑线性模型丁“Y一召…月二‘L刀Y一u”V,V>o,O’“为简便计,记为之Y,了月,砂V,V>0).若召是:x,阵,伺题.未知当习月可估时, (1)我们研究估计尽月的 RaoL刀给出了,在二次型损失函数 (‘一习月丫(d一习月)(2)下,S月的齐次线性估计L了在齐次线性估计类中是可容许估计的充要条件.本文考虑月月的非齐次线性估计L了+a的可容许性.在二次型损失函数下得到了LY十a在非齐次线性估计类中是习月的可容许估计的充要条件. 称R(S月,。,,d、~E(‘一习月)‘(d一召月)为习月的估计d的风险.若d,,d。是S尽的两个估计,当R〔泞月,。,,J,)一R(习月,a,,d:…     

矩阵损失函数下回归系数矩阵的非齐次线性估计的可容许性

在二次矩阵损失函数下研究了协方差矩阵未知的多元线性模型中回归系数矩阵的可估线性函数的矩阵非齐次线性估计的可容许性,给出了矩阵非齐次线性估计在线性估计类中可容许的一个充要条件.     

多元线性模型中回归系数矩阵非齐次线性估计的可容许性

该文研究了协方差矩阵未知的多元线性模型中,二次矩阵损失函数下回归系数矩阵可估线性函数的非齐次线性估计的可容许性.不需正态分布的假设,作者给出矩阵非齐次线性估计在线性估计类中可容许的充要条件;在正态分布的假设下,作者给出矩阵非齐次线性估计在一切估计组成的估计类中可容许的充分条件.     

具有特殊协方差结构的 SURE 模型中参数估计的若干结果

本文讨论具有特殊协方差结构似乎不相关回归方程（SURE）模型中参数的估计问题．除非另有说明,损失函数将取为二次损失和矩阵损失．本文证明了回归系数的线性可估函数的最小二乘估计是极小极大的且在矩阵损失函数下是可容许的;还分别在仿射交换群和平移群下导出了存在回归系数的线性可估函数的一致最小风险同变（UMRE）估计的充要条件,并证明了在仿射交换和二次损失下不存在协方差阵和方差的UMRE估计．     

Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models. The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given.     

Improved estimation of a multinormal precision matrix

Stein's technique is used to obtain improved estimators of the multinormal precision matrix under quadratic loss. The technique is to obtain a certain differential inequality involving the eigenvalues of the sample covariance matrix. Several improved estimators are obtained by solving the differential inequality.     

Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss

The problem of estimating large covariance matrices of multivariate real normal and complex normal distributions is considered when the dimension of the variables is larger than the number of samples. The Stein–Haff identities and calculus on eigenstructure for singular Wishart matrices are developed for real and complex cases, respectively. By using these techniques, the unbiased risk estimates for certain classes of estimators for the population covariance matrices under invariant quadratic loss functions are obtained for real and complex cases, respectively. Based on the unbiased risk estimates, shrinkage estimators which are counterparts of the estimators due to Haff [L.R. Haff, Empirical Bayes estimation of the multivariate normal covariance matrix, Ann. Statist. 8 (1980) 586–697] are shown to improve upon the best scalar multiple of the empirical covariance matrix under the invariant quadratic loss functions for both real and complex multivariate normal distributions in the situation where the dimension of the variables is larger than the number of samples.     

On the quadratic estimation of covariance matrices in multivariate linear models

This paper investigates the estimation of covariance matrices in multivariate mixed models. Some sufficient conditions are derived for a multivariate quadratic form and a linear combination of multivariate quadratic forms to be the BQUE (quadratic unbiased and severally minimum varianced) estimators of its expectations.     

On improved estimation of normal precision matrix and discriminant coefficients

The problem of estimating the precision matrix of a multivariate normal distribution model is considered with respect to a quadratic loss function. A number of covariance estimators originally intended for a variety of loss functions are adapted so as to obtain alternative estimators of the precision matrix. It is shown that the alternative estimators have analytically smaller risks than the unbiased estimator of the precision matrix. Through numerical studies of risk values, it is shown that the new estimators have substantial reduction in risk. In addition, we consider the problem of the estimation of discriminant coefficients, which arises in linear discriminant analysis when Fisher's linear discriminant function is viewed as the posterior log-odds under the assumption that two classes differ in mean but have a common covariance matrix. The above method is also adapted for this problem in order to obtain improved estimators of the discriminant coefficients under the quadratic loss function. Furthermore, a numerical study is undertaken to compare the properties of a collection of alternatives to the “unbiased” estimator of the discriminant coefficients.     

Discriminant analysis for compositional data and robust parameter estimation

Compositional data, i.e. data including only relative information, need to be transformed prior to applying the standard discriminant analysis methods that are designed for the Euclidean space. Here it is investigated for linear, quadratic, and Fisher discriminant analysis, which of the transformations lead to invariance of the resulting discriminant rules. Moreover, it is shown that for robust parameter estimation not only an appropriate transformation, but also affine equivariant estimators of location and covariance are needed. An example and simulated data demonstrate the effects of working in an inappropriate space for discriminant analysis.     

Methods for improvement in estimation of a normal mean matrix

This paper is concerned with the problem of estimating a matrix of means in multivariate normal distributions with an unknown covariance matrix under invariant quadratic loss. It is first shown that the modified Efron-Morris estimator is characterized as a certain empirical Bayes estimator. This estimator modifies the crude Efron-Morris estimator by adding a scalar shrinkage term. It is next shown that the idea of this modification provides a general method for improvement of estimators, which results in the further improvement on several minimax estimators. As a new method for improvement, an adaptive combination of the modified Stein and the James-Stein estimators is also proposed and is shown to be minimax. Through Monte Carlo studies of the risk behaviors, it is numerically shown that the proposed, combined estimator inherits the nice risk properties of both individual estimators and thus it has a very favorable risk behavior in a small sample case. Finally, the application to a two-way layout MANOVA model with interactions is discussed.