Nonparametric Spline Estimation On Smoothers In Semiparametric Models With Longisudinal Data

In recent years , a lot of econometric literatures have been devoted to non-parametric estimation in semiparametric models , there are profile-kernel method and backfitting method as so on smoothing. This paper considers smoothing spline method with longitudinal data , supposes that there are n subjects and for the i-th subject, there are n₁ observations. The model is Y_ij = X_ij^T β + g(t_i) +ε_i(t_ij), where (Y_ij,X_ij,t_ij) are observations , β = (β₀, … ,βk)^T,g{t) is arbitrary smooth function of t , and the error term ε_i(t) is a zero-mean , identically stochastic process. The observations are assumed to be independent for different subjects , β and g(t) are needs to be estimated.This paper presents the spline estimation and algrithm for every individual which has the same or different observation . And then we consider the nonparamet-ric function g(t) , when it's a monotone increasing function , we get the expression of the estimation . For the smoothing parametric , we give a cross-validation rule.