强收敛意义下岭回归的 C_L 和 GCV 准则的渐近最优性

ASYMPTOTIC OPTIMALITY OF RIDGE PARAMETERS BASED ON C_L AND GCV PROCEDURES WITH RESPECT TO STRONG CONVERGENCE

设有回归模型Y_i=μ_i+e_i,i=1,2,…,n (1)假定 e_1,…,e_n 为 iid.的正态随机变量序列,具有共同的均值0和方差σ~2.每个 Y_i 可通过设计点列 x_(i1),x_(i2),…,x_i_p_n 观察到.为估计 Y=(Y_1,…,Y_n)′的未知均值 μ=(μ_1,…,μ_n)′,可构造一族岭估计(?)(h)=X(X′X+hI)^-1X′Y,h≥0,(2)其中 X=(x_ij)_(n×ρn) 为设计阵,I 为 p_n 阶单位阵.在这里,岭参数 h

Consider regression modelY_i=μ_i+e_i,i=1,…,n,where e_1,…,e_n are i.i.d.random errors,with common distribution N(0, σ~2).Forthe estimation of mean vector μ=(μ_1,…,μ_n)′,ridge estimators (?)_n(h)=X(X′X+hl)^(-1)X′Y,h≥0,are to be constructed on the basis of design matrix X=(x_(ij))_(n×p_n).In this paper,it is proved that the ridge parameters (?)_M and (?)_Q based on C_L and GCVprocedures,respectively,are both asymptotically optimal with respect to strong con-vergence,i.e.,for (?)=(?)_M and (?)_Q,‖u-(?)_n((?))‖~2/(?)‖μ-(?)_n(h)‖~2→1,with probability one.