Conditional Linear Minimax Predictor In The Finite Population With Linear Equality Constraint

The Minimax predictor on the general linear model is the problem which was studiedby many scholars all along. Many perfect results had been obtained in this field.In this paper, we consider the conditional linear modelwhereεis a n×1 random error vector,βis unknown p-dimensional parameter,σ² > 0is parameter and V≥0, H is k×p constraint matrix. Consider the conditional linearMinimax predictor of the predictable variable.Firstly, under quadratic loss function, the conditional linear model is changed to un-conditional linear model usingβ= (I ?H+H)γ= NHγ. Assuming that V_sX_sN_HX_s^T (I -V_s⁺ V_s) = 0 and raising a homogeneous linear predictor according to (L-A_s)Q₂Q₂^TX_sN_H =T₂Q₂^TX_sN_H, the sufficient and necessary condition of the entity of the Minimax predic-tor of Ay is put forward. Then, the conditional linear Minimax predictor in the classof homogeneous linear predictor is given by calculating and simplification and is provedunique in the finite population with arbitrary rank.Secondly, under quadratic loss function and the same hypotheses, let the randomerror obey normal distribution, that isε～N(0,σ²V ). At this point, using singular valuedecomposition of the matrix, the conditional linear Minimax predictor in the class of allpredictors is obtained through a direct calculation in the finite population with arbitraryrank and using the conclusion about admissibility, the uniqueness is proved.Thirdly, under the matrix loss function, the necessary condition of that M be aupper bound of venture is obtained and the unique partial conditional linear Minimaxpredictor in the class of linear predictor is obtained in the finite population with arbitraryrank in the case of Q₂^TX_sG^- = 0, Q₁^T X_sN^- = 0 and 0 <σ²â‰¤Î²^TX_s^TV+s + X_sβ.Finally, we change the multivariate linear regression model to the single linear re-gression model on the base of the vector of matrix, then use the conclusion obtained fromthe former chapter, the conditional linear Minimax predictor is obtained under quadraticloss function and the matrix loss function in the multivariate linear regression model.