  The complexity of semiparametric models poses new challenges to statistical
inference and model selection that frequently arise from real applications. In
this work, we propose new estimation and variable selection procedures for the
semiparametric varying-coefficient partially linear model. We first study
quantile regression estimates for the nonparametric varying-coefficient
functions and the parametric regression coefficients. To achieve nice
efficiency properties, we further develop a semiparametric composite quantile
regression procedure. We establish the asymptotic normality of proposed
estimators for both the parametric and nonparametric parts and show that the
estimators achieve the best convergence rate. Moreover, we show that the
proposed method is much more efficient than the least-squares-based method for
many non-normal errors and that it only loses a small amount of efficiency for
normal errors. In addition, it is shown that the loss in efficiency is at most
11.1% for estimating varying coefficient functions and is no greater than 13.6%
for estimating parametric components. To achieve sparsity with high-dimensional
covariates, we propose adaptive penalization methods for variable selection in
the semiparametric varying-coefficient partially linear model and prove that
the methods possess the oracle property. Extensive Monte Carlo simulation
studies are conducted to examine the finite-sample performance of the proposed
procedures. Finally, we apply the new methods to analyze the plasma
beta-carotene level data.
