Asymptotic normality of M-estimators in a semiparametric model with longitudinal data

摘要

Suppose that the longitudinal observations (Y-ij, X-ij, t(ij)) for i = 1, ... , n; j = 1, ... , m(i) are modeled by the semiparamtric model Y-ij = X-ij(T)beta(0) + g(t(ij)) + e(ij), where beta(0) is a k x 1 vector of unknown parameters, g(.) is an unknown estimated function and e(ij) are unobserved disturbances. This article consider M-type regressions which include mean, median and quantile regressions. The M-estimator of the slope parameter beta(0) is obtained through piecewise local polynomial approximation of the nonparametric component. The local M-estimator of g(.) is also obtained by replacing beta(0) in model with its M-estimator and using local linear approximation. The asymptotic distribution of the estimator of beta(0) is derived. The asymptotic distributions of the local M-estimators of g(.) at both interior and boundary points are also established. Various applications of our main results are given.