任意秩多元线性模型中的简单投影预测

The simple projection predictor in multivariate linear model with arbitrary rank

考虑任意秩多元线性模型Y =XB +ε(其中 ,E(Vec(ε) ) =0 ,V(Vec(ε) ) =σ2 Δ Σ) ,该模型的预测问题就是利用已观察值矩阵Y预测未观察值矩阵Y0 =X0 B +ε0 .H .Bolfarine等强调预测必须有简洁而直观的形式 ,最简洁、直观的预测是简单投影预测 (SPP) ;对于超总体模型 ,H .Bolfarine等得到了简单投影预测为最优预测的充要条件 .作者研究了预测的最优性 ,对任一线性可预测变量θ =KY0 L ,它的简单投影预测被定义为 ^θSPP =KX0 (X′T- X) - X′T- YL(其中 ,T =Σ +XX′) ;得到了 ^θSPP为θ的最优线性无偏预测的充要条件 ,并研究了 ^θSPP关于协方差矩阵的稳健性 ,从而推广了H .

Considering the multivariate linear model with arbitrary rank Y=XB+ε ,where E( Vec (ε))=0 and V( Vec (ε))=σ 2ΔΣ, the unknown observation matrix Y 0=X 0B+ε 0 is predicted using the known observation matrix Y .Srndal and Wright emphasized the need for predictors with simple and intuitive forms.The most intuitive and simple predictor certainly is the simple projection predictor (SPP). H.Bolfarine, et al obtained several necessary and sufficient conditions for optimum of the simple projection predictor under a superpopulation model. The above multivariate linear model is studied. For arbitrary linear predictable variable θ=KY 0L ,its SPP is then defined by SPP = KX 0(X ′T -X) -X ′T -YL, where T=Σ+XX ′ .A number of necessary and sufficient conditions where SPP is also the best linear unbiased predictor are obtained,and the robustness of the SPP on the covariance matrix is investigated, and thus the relevant results drawn by H. Bolfarine, et al are widely used.