  This paper is concerned with a semiparametric partially linear regression
model with unknown regression coefficients, an unknown nonparametric function
for the non-linear component, and unobservable Gaussian distributed random
errors. We present a wavelet thresholding based estimation procedure to
estimate the components of the partial linear model by establishing a
connection between an $l_1$-penalty based wavelet estimator of the
nonparametric component and Huber's M-estimation of a standard linear model
with outliers. Some general results on the large sample properties of the
estimates of both the parametric and the nonparametric part of the model are
established. Simulations and a real example are used to illustrate the general
results and to compare the proposed methodology with other methods available in
the recent literature.
