L1-estimation in a semiparametric model with longitudinal data

Consider the semiparametric model Y ij = X i T β 0 + g ( t ij ) + e ij , where β 0 is a k × 1 vector of unknown parameters, g ( · ) is a function to be estimated and e ij are unobserved disturbances. A piecewise polynomial is proposed to approximate g and two least absolute deviation estimators of β 0 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 ( · ) are also obtained by replacing β 0 in this model with their least absolute deviation estimators and using a local linear approximation. The asymptotic distributions of the estimators of β 0 are derived. The asymptotic distributions of the local least absolute deviation estimators of g ( · ) at both interior and boundary points are also established. Finite sample properties of our procedures are studied through Monte Carlo simulations.