一类半参数回归模型中M估计的收敛速度

Convergence rates of M-estimates in a semiparametric regression model

考虑半参数回归模型yi=xTiβ0+g(ti)+ei,i=1,2,…,n。其中,β0是未知参数,g是未知函数。当g的估计取一类非参数权估计(包括核估计和最近邻估计)时,文章讨论了参数β0的M估计β0的强收敛速度和未知函数g的估计g*n(t)的一致强收敛速度,从而得到β0-β0=O(n-1/2(logn)1/2) a.s.和sup|g*n(t)-g(t)|=O(n1/3logn) a.s.。

Here considered is the semiparametric regression model:y_i=x^Tiβ_0+g(t_i)+e_i,i=1,2,…,n, where β_0 is an unknown parameter and g an unknown function. The strong convergence rate for M-estimates _0 of the parameter β_0 and the uniform strong convergence rate for the estimate ~*_n(t) of function g which is a class of nonparametric weight estimates(including the kernel estimate and the nearest neighbour estimate) are discussed.The results are obtained as follows:_0-β_0=O(n^(-1/2)(logn)^(1/2)) a.s.and (sup)0≤t≤1｜~*_n(t)-g(t)｜=O(n^(1/3)logn)a.s.