Wavelet estimation of partially linear models

Abstract

A wavelet approach is presented for estimating a partially linear model (PLM). We find an estimator of the PLM by minimizing the square of the l 2 norm of the residual vector while penalizing the l 1 norm of the wavelet coefficients of the nonparametric component. This approach, an extension of the wavelet approach for nonparametric regression problems, avoids the restrictive smoothness requirements for the nonparametric function of the traditional smoothing approaches for PLM, such as smoothing spline, kernel and piecewise polynomial methods. To solve the optimization problem, an efficient descent algorithm with an exact line search is presented. Simulation results are given to demonstrate effectiveness of our method.