矩阵损失下的协方差阵Σ的二次型估计的风险函数

The risk function of the quadratic estimate for the covariance matrix Σ under a matrix loss function

摘要: 考虑模型 H:Y=( Y1 ,Y2 ,… ,Yn)′=( X′1 ,X′2 ,… ,X′n)′β+ ( e1 ,e2 ,… ,en)′ Xβ+ e.其中 ,Yi:r维列观察向量 ,Xi:r× p已知矩阵 ,i=1 ,2 ,… ,n.β=( β1 ,β2 ,… ,βp)′是 p维未知参数向量 .e1 ,e2 ,… ,en　iid,e1 与r维正态分布 Nr( 0 ,Σ)有相同的前 4阶矩 ,这里Σ是未知的 r× r协方差阵 .在矩阵损失函数 L( d,Σ) =( d-Σ) 2 下 ,给出了Σ的二次型估计类 Y′AY:A≥ 0 ,A∈ Rn× n的风险函数 .

Abstract: Considering the model H:Y(Y 1,Y 2,...,Y n)′=(X′ 1,X′ 2,...,X′ n)′β+(e 1,e 2,...,e n)′Xβ+e.Where,Y i is an r-dimensional observation vector,X i is a r×p known matrix,i=1,2,...,n.β(β 1,β 2,...,β p)′is the p dimensional unknown parameter vector.e 1,e 2,...,e n iid,e 1 has the same first four moments as N r(0,Σ),Σ is an r×r unknown covariance matrix.Under matrix loss function L(d,Σ)=(d-Σ) 2,the risk function of the quadratic estimate Y′AY (A≥0) for Σ is given.