MANOVA模型中均值参数的极大极小估计和可容许估计

设Y_(n×m)服从矩阵正态分布N(XΘ, σ^2Σ×V),X_(n×k)是一个列满秩的矩阵,n ≥k≥3,σ^2是未知的,σ^(-2)S_p服从自由度为p的χ^2分布。当f(t)是单调非降 可微的函数, 且0≤f(t)≤2(k-2)/m(p+2)时,其列向量为Δ_i(Y)=I_k- f(V′_iY′(X′Σ^(-1)X)^(-2)YV_iS_p_(-1)S_p (X′Σ^(-1)X)^(-1)/V′_iY′(X′Σ^(-1)X)^(-2)YV_i](X′Σ^(-1)X)^(-1)X′Σ^(-1)Y_i的估计Δ(Y)在风险函数R_1或R_2下是能够改善Θ的极大似然估计(X′Σ^(-1)X)^(-1)X′Σ^(-1)Y.同时得到了Θ和CXΘ的线性可容许估计类. 

Minimax Estimation and Admissible Estimation Under Normal MANOVA Model

Let random matrix Y_(n×m) be distributed according to N(XΘ, σ^2Σ×V),Where X is a known n×k matrix of rank k, n≥k≥3, σ^2 is unknown  and σ^(-2)S_p has a χ square distribution with degree of freedom p. When f(t) is differentiable and f′(t)≥0,0≤f(t)≤2(k-2)/m(p+2), the estimator whose ithcolumn vector has the formΔ_i(Y)=I_k- f(V′_iY′(X′Σ^(-1)X)^(-2)YV_iS_p_(-1)S_p (X′Σ^(-1)X)^(-1) /V′_iY′(X′Σ^(-1)X)^(-2)YV_i](X′Σ^(-1)X)^(-1)X′Σ^(-1)Y_i where V_i is the ith column vector of  V^(-1/2),can improve the maximum likelihood estimator (X′Σ^(-1)X)^(-1)X′Σ^(-1)Y of Θ. Moreover, some  classes  of admissible linear estimators for Θ or  CXΘ  are obta ined too