L1L1-estimation in a semiparametric model with longitudinal data

Consider the semiparametric model Yij=XiTβ0+g(tij)+eij, where β0β0 is a k×1k×1 vector of unknown parameters, g(·)g(·) is a function to be estimated and eijeij are unobserved disturbances. A piecewise polynomial is proposed to approximate g   and two least absolute deviation estimators of β0β0 are obtained by using two weighting schemes: equal weight for each subject and equal weight for each measurement. Two local least absolute deviation estimators of g(·)g(·) are also obtained by replacing β0β0 in this model with their least absolute deviation estimators and using a local linear approximation. The asymptotic distributions of the estimators of β0β0 are derived. The asymptotic distributions of the local least absolute deviation estimators of g(·)g(·) at both interior and boundary points are also established. Finite sample properties of our procedures are studied through Monte Carlo simulations.