矩阵损失下线性模型中参数的最优同变估计

The Best Invariance Estimitors of the Parameters for the Linear Model Under the Matrix Loss Function

本文考虑线性模型:Y=Xβ+ε,其中X为n×p阶矩阵,rank(X)=r≤p,n维随机误差向量ε～F(z/σ),σ>0,参数β∈R^p。当σ已知时,在平移交换群和损失函数(d(Y)-Qβ)(d(Y)-Qβ)′下,作者给出了β的线性可估函数Qβ(其中Q为k×p阶矩阵)的最优同变估计;当σ未知时,在仿射变换群和损失函数 (b_1σ^(-1)I_k 0 0 b_2σ^(-q))(d_1(Y)-Qβ d_2(Y)-σ~q)(d_1(Y)-Qβ d_2(Y)-σ~q)′(b_1σ^(-1)I_k 0 0 b_2σ^(-q)之下,给出了(Qβ σ~q)的最优同变估计,其中q>0,b_1>0,b_2≥0。

Consider the linear model Y = Xβ + ε, where X be a n × p known design matrix, rank(X) = r ≤ p, ε be a n-demensional random error vector ε- F(Z/σ), σ > 0,β be a parameter vector and β∈ Rp. In this paper, when σ is known, we give the best invariant estimator of Qβ (Q be a n × p matrix and Qβ be the linear estimator functin of β) under the translation group and the matrix loss function (d(Y) - Qβ)(d(Y) - Qβ)'; when σ is unknown,we give the best invariant estimator of under the affine group and the loss functionwhere q > 0, b1 > 0, b2≥0.