The Linear Minimax Estimator of Mean Vector in Multivariate Statistics under Matrix Loss

Let y₁，y₂，…，y_n be i.i.d., Ey₁＝β,Cov（y₁）=∑, where β∈R^P and ∑ ＞ 0 are unknown. We estimate β The class of estimation is L=｛∑_i=1ⁿ L_iy_i:L_i is a p-order constant matrix, i =1, 2,…, n. The loss function is L_V_1, V_2(\beta, \Sigma ; d)=\frac(d-\beta)(d-\beta)^\prime\operatornametr\left(q \Sigma V_1\right)+\beta^\prime V_2 \beta, whereV₁,V₂ are known. We study the minimaxity of a linear estimator of β inL .The main results are 1. When V₂＝kV₂，k＞0, the only Ⅰ-type linear minimax estimator of β inL is Y/（1＋√k ）,where Y＝\barY＝\frac1n \sum_i=1^n y_i.2. When V₂＝kV₂ is failed for all k ＞0, but V₁V₂＝V₂V₁, the Ⅰ-type linear minimax estimator of β in L dosn’t exist.3. When V₁V₂＝V₂V₁, the Ⅱ-type linear minimax estimator of β in L is(V₁^1/2+V₂^1/2)^-1V₁^1/2Y .These estimators constitute a set AY:A is symmatric and all latent roots of A belong to （0, l）when both V₁ and V₂ vary under subjection of V₁V₂＝V₂V₁.4. For general V₁ and V₂,(V₁^1/2+V₂^1/2)^-1V₁^1/2Y, is also a Ⅱ-type linear minimax estimator of β in L. These estimators constitute a set AY: all latent roots of A are real and belong to （0,1）, and A only has linear element factors when both V₁ and V₂ vary