参数受约束时线性模型的线性Minimax估计

Minimax Linear Estimation in a Linear Model with Parameter Restriction

考虑线性模型Y=β+ε,Eε=0,Var(ε)=σ~2V,其中Y为n维观察向量,V≥0为已知n×n矩阵,β∈R^n,σ~2>0为未知参数,受椭球约束U={β:β′Hβ≤σ~2},H为已知n×n矩阵,C.R.Rao在[1]中给出了H>0时,p′β的唯一线性minimax估计,本文讨论了一般的非负定矩阵H≥0,H≠0的情况,也给出了p′β的唯一的线性minimax估计.

In consideration of the linear model Y = β+ε, Eε=0, Var(ε)==σ2V, where V≥0 is known n. n. d matrix, β∈Rn and σ2>0 are parameters, Let U={β: β'Hβ≤σ2}, where H is a n.n.d matrix. The unque minimax estimater of p'β is q*'Y in the sense of min max E(q'Y - p'β)2 = max E(q*'Y - p'β)2> where q* = [I-H(HVH+H,)+HV]p.