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

Abstract

Suppose that the longitudinal observations (Yij, Xij, tij) for i = 1, . . . ,n; j = 1, . . . ,mi are modeled by the semiparamtric model \({Y_{ij}=X_{ij}^{T}\beta_{0}+g(t_{ij})+e_{ij},}\) where β0 is a k × 1 vector of unknown parameters, g(·) is an unknown estimated function and eij are unobserved disturbances. This article consider M-type regressions which include mean, median and quantile regressions. The M-estimator of the slope parameter β0 is obtained through piecewise local polynomial approximation of the nonparametric component. The local M-estimator of g(·) is also obtained by replacing β0 in model with its M-estimator and using local linear approximation. The asymptotic distribution of the estimator of β0 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.