PC准则下回归系数的一类线性估计的优良性

The Superiority about a Class of Linear Estimaton of Regression Coefficient under Pitman Closeness Criterion

设线性回归模型为,此处n≥p,X的秩为R(X)=s,0<s≤p;令回归系敷的最小二乘(LS)解和一类线性估计分别为和,其中p>0为常数,∑_0为正定阵。本文证明了:在适当条件下(?)于PC准则下优于(?)并将这一结果应用于回归系数的岭估计、广义岭估计、压缩估计和Bayes估计。

Let the linear regression model be Ynxi = Xnxpβp×1+εn×1, where n ≥ p, rank(X) = s, and ε～N(0,σ2). Suppose that the LS solution and linear estimation of regression coefficient are β= (X'X)-X'Y and βp = (X'X +pΣ0 )-1X'Y, where p > 0 is a contant and Σ0 is a positive definite matrix. In this paper we prove that under suitable conditions the linear estimator βp is better than βby Pitman closeness criterion, and apply this result to the ridge estimators, generalized ridge estimators, shrinkage estimators and Bayes estimators.