Existence of the uniformly minimum risk equivariant estimators of parameters in a class of normal linear models

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrixV and (trV)^α, where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators ofV and tr(V) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.     

一类线性模型下最优线性无偏估计的存在性

考虑包含如方差分量模型、似乎不相关回归方程模型、增长曲线模型和扩充的增长曲线模型等众多常见模型的一类较广泛的线性模型。对模型中误差向量的分布不作假定时,给出了在二次损失或矩阵损失下存在回归系数的线性可估函数的一致最小风险线性无偏估计的充分必要条件。     

误差服从多元t分布的一类线性模型下参数估计的若干注记

研究一类线性模型下参数估计的若干问题．这类模型包含了多个因变量线性模型、增长曲线模型、扩充的增长曲线模型、似乎不相关回归方程组、方差分量模型等常用模型．在这类线性模型下,证明了当误差服从多元t分布时与误差服从多元正态分布时,具有相同的完全统计量和无偏估计,且在后一种情况下的充分统计量必为前一种情况下的充分统计量．对于带有多种协方差结构的前述几种模型,把在误差服从多元正态分布下,相应的协方差阵及有关参数的一致最小风险无偏(UMRU)估计存在性的结论推广到了相应的误差服从多元t分布情形．此外,对于误差服从多元t分布的这类统一的线性模型,给出了回归系数的线性可估函数的无偏估计的协方差阵的C-R下界．     

增长曲线模型中UMRE估计和UMRU估计的存在性

对于一般的增长曲线模型,在一般的矩阵损失和二次损失下,用统一的方法分别给出了回归系数矩阵的任一指定可估函数存在一致最小风险同变（ＵＭＲＥ）估计（分别在仿真变换群和转换变换群下）和一致最小风险无编（ＵＭＲＵ）估计的充要条件,以及所有可估函数恒存在ＵＭＲＥ估计和ＵＭＲＵ估计的允要条件。最后将结果应用于一些特殊模型。     

Admissibilities of linear estimator in a class of linear models with a multivariate t error variable

This paper discusses admissibilities of estimators in a class of linear models,which include the following common models:the univariate and multivariate linear models,the growth curve model,the extended growth curve model,the seemingly unrelated regression equations,the variance components model,and so on.It is proved that admissible estimators of functions of the regression coefficient β in the class of linear models with multivariate t error terms,called as Model II,are also ones in the case that error terms have multivariate normal distribution under a strictly convex loss function or a matrix loss function.It is also proved under Model II that the usual estimators of β are admissible for p 2 with a quadratic loss function,and are admissible for any p with a matrix loss function,where p is the dimension of β.     

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．     

Existence of the uniformly minimum risk unbiased estimator in seemingly unrelated regression system

For a seemingly unrelated regression system with the assumption of normality, a necessary and sufficient condition for the existence of the Uniformly Minimum Risk Unbiased (UMRU) estimator of regression coefficients under strictly convex loss is obtained; it is proved that any unbiased estimator can not improve the least squares estimator; it is also shown that no UMRU estimator exists under missing observations.Supported by the National Natural Science Foundation of China.     

带有结构变化的线性模型中参数估计的一些结果

本文在一些纯量损失和矩阵损失下研究带有结构变化的正态线性模型中参数的估计问题．分别给出 了存在回归系数的一致最小风险无偏（ＵＭＲＵ）估计和一致最小风险同变（ＵＭＲＥ）估计的充要条件, 证明了不存在误差方差在仿射变换群下的ＵＭＲＥ估计．导出了回归系数的最小二乘估计的可容许性 和极小极大性．     

All admissible linear estimators of a regression coefficient under a balanced loss function

Admissibility of linear estimators of a regression coefficient in linear models with and without the assumption that the underlying distribution is normal is discussed under a balanced loss function. In the non-normal case, a necessary and sufficient condition is given for linear estimators to be admissible in the space of homogeneous linear estimators. In the normal case, a sufficient condition is provided for restricted linear estimators to be admissible in the space of all estimators having finite risks under the balanced loss function. Furthermore, the sufficient condition is proved to be necessary in the normal case if additional conditions are assumed.     

带约束生长曲线模型线性估计的泛容许性

本文讨论带约束生长曲线模型中回归系数线性估计的泛容许性,给出了回归系数的线性估计在线性估计类中是泛容许估计的充要条件。     

Difference based ridge and Liu type estimators in semiparametric regression models

We consider a difference based ridge regression estimator and a Liu type estimator of the regression parameters in the partial linear semiparametric regression model, y=Xβ+f+ε. Both estimators are analyzed and compared in the sense of mean-squared error. We consider the case of independent errors with equal variance and give conditions under which the proposed estimators are superior to the unbiased difference based estimation technique. We extend the results to account for heteroscedasticity and autocovariance in the error terms. Finally, we illustrate the performance of these estimators with an application to the determinants of electricity consumption in Germany.     

ADMISSIBILITY OF LINEAR ESTIMATORS IN A GROWTH CURVE MODEL SUBJECT TO AN INCOMPLETE ELLIPSOIDAL RESTRICTION

This article considers the admissibility of the linear estimators for the regression coefficients in the growth curve model subject to an incomplete ellipsoidal restriction. The necessary and sufficient conditions for linear estimators to be admissible in classes of the homogeneous and non-homogeneous linear estimators, respectively, are obtained under the quadratic loss function. They are generalizations of some existing results in literature.     

增长曲线模型回归系数线性估计的可容许性

该文讨论增长曲线模型中回归系数线性估计的可容许性, 在矩阵损失(d(Y)-KBL)(d(Y)-KBL)' 下, 分别给出了回归系数的齐次与非齐次线性估计是可容许的充要条件, 推广了以往文献的相关结论.     

Admissibility of linear estimators with respect to inequality constraints under matrix loss function

In this paper we investigate the admissibility of linear estimators in the multivariate linear model with respect to inequality constraints under matrix loss function. The necessary and sufficient conditions for a linear estimator to be admissible in the class of homogeneous linear estimators and the class of inhomogeneous linear estimators are obtained, respectively.     

On the equivalence of estimations under a general linear model and its transformed models

Some necessary and sufficient conditions are given for two equalities of ordinary least-squares estimators and best linear unbiased estimators of an estimable vector of parametric functions under a general linear model and its transformed linear model to hold     

ADMISSIBILITY IN THE GENERAL GROWTH CURVE MODEL WITH RESPECT TO RESTRICTED PARAMETER SETS UNDER MATRIX LOSS FUNCTION

In this paper, we study the issue of admissibility in the growth curve model with respect to restricted parameter sets under matrix loss function. We obtain some neces- sary and sufficient conditions that the linear estimators of KBL are admissible in the class of homogeneous linear estimators and in the class of non-homogeneous linear estimators under the growth curve model with respect to restricted parameter sets, respectively.     

Statistical inference of partially linear regression models with heteroscedastic errors

The authors study a heteroscedastic partially linear regression model and develop an inferential procedure for it. This includes a test of heteroscedasticity, a two-step estimator of the heteroscedastic variance function, semiparametric generalized least-squares estimators of the parametric and nonparametric components of the model, and a bootstrap goodness of fit test to see whether the nonparametric component can be parametrized.     

Functional-coefficient partially linear regression model

In this paper, the functional-coefficient partially linear regression (FCPLR) model is proposed by combining nonparametric and functional-coefficient regression (FCR) model. It includes the FCR model and the nonparametric regression (NPR) model as its special cases. It is also a generalization of the partially linear regression (PLR) model obtained by replacing the parameters in the PLR model with some functions of the covariates. The local linear technique and the integrated method are employed to give initial estimators of all functions in the FCPLR model. These initial estimators are asymptotically normal. The initial estimator of the constant part function shares the same bias as the local linear estimator of this function in the univariate nonparametric model, but the variance of the former is bigger than that of the latter. Similarly, initial estimators of every coefficient function share the same bias as the local linear estimates in the univariate FCR model, but the variance of the former is bigger than that of the latter. To decrease the variance of the initial estimates, a one-step back-fitting technique is used to obtain the improved estimators of all functions. The improved estimator of the constant part function has the same asymptotic normality property as the local linear nonparametric regression for univariate data. The improved estimators of the coefficient functions have the same asymptotic normality properties as the local linear estimates in FCR model. The bandwidths and the smoothing variables are selected by a data-driven method. Both simulated and real data examples related to nonlinear time series modeling are used to illustrate the applications of the FCPLR model.     

Special variance structures in the growth curve model

Standard and extended growth curve model (multivariate linear model) with practically important variance structures are considered and a method for parameters estimation is proposed.     

A subclass of bayes linear estimators that are minimax

In the linear regression model with ellipsoidal parameter constraints, the problem of estimating the unknown parameter vector is studied. A well-described subclass of Bayes linear estimators is proposed in the paper. It is shown that for each member of this subclass, a generalized quadratic risk function exists so that the estimator is minimax. Moreover, some of the proposed Bayes linear estimators are admissible with respect to all possible generalized quadratic risks. Also, a necessary and sufficient condition is given to ensure that the considered Bayes linear estimator improves the least squares estimator over the whole ellipsoid whatever generalized risk function is chosen.