多元线性模型中未知参数阵的同时估计的可容许性

Admissibility of Estimators of Unknown Parameter Matrix in A Multivariate Linear Model

本文考虑多元线性模型 Y=χβ′+εΕε=0 cov(vec(ε)=Σ(×)V (1.1)其中Y是N×P阶矩阵,x和V分别是N×1和N×N阶已知矩阵(V>0.x≠0)。Σ和β分别是P×P和p×1阶参数矩阵(Σ>0).ε=V^(1/2)ZΣ^(1/2)、Z=(z_(ij))(i=1.2.….N;j=1.2.…p).z_(ij)相互独立,E(z_(ij))=O.E(z_(ij)~2)=1.E(z_(ij)~3)=0.E(z_(ij)~4)=3.vec(ε)表示将ε按列拉直.Σ(×)V表示Σ与V的kroneker积,我们得到了下 N>P面的主要结果: 1°在模型(1.1)和损失函数 tr(d_1-β)Σ^(-1)(d_1-β′)′+ tr(d_2Σ^(-1)-I)~2之下,给出了(β′Σ)的同时估计(l′Y.Y′AY)在估计类γ_1+{((l′Y,Y′AY):l为N维向量,A≥0}中是可容许的充要条件。 2°在模型(1.1)和损失函数 tr(d_1-β′)Σ^(-1)(d′-β′)′+tr(d_2Σ^(-1)-I-βx′xβ′Σ^(-1))~2之下,给出了(β′.Σ+βx′xβ)的同时估计(l′Y.Y′AY)在估计类γ_1中是可容许的充要条件。

In this paper we consider the linear model Y=xβ′+ε Eε=0 cov(vec(ε))=∑(×)V where y is a Random matrix of dimension N×P Both x_(N×1)≠0 and V_(N×N)>0 one giv- en. Both ∑_(p×p) and β_(p×1) are unknown. ε=V^(1/2)ZV^(1/2), Z=(z_(ij)) (i=1. 2.…N:j=1. 2. …p). z_(ij) are independent. E(z_(ij))=E(z_(ij)~3)=0, E(z_(ij)~2)=1, E(z_(ij)~4)=3. Let E_(N×p)=(ε_1. ε_2…. ε_p). denote vec (ε′_1. ε′_2…. ε′_p )′. ∑(×)V denotes the kroneker product of ∑ and V. We get the following main results: the necessary and sufficient conditions that (l′Y. Y′AY) is an admissible estimetor for (β′. ∑) with in the class _1={ (l′Y,Y′AY): L∈R^N. A≥0} under the loss function tr (d_1- β~1) ∑^(-1) (d_1-β~1)′+tr (d_2 ∑^(-1)-I)~2 are given. The necessary and sufficient conditions that (l′Y. Y′AY) is an admissible estimator for (β′, ∑+βx′xβ) with in the class _1 under the loss function tr(d_1-β~1)∑^(-1)(d_1-β~1)′+tr(d_2∑^(-1)-I-βX′Xβ′∑^(-1))~2 are given.