  In this paper, we are concerned with how to select significant variables in
semiparametric modeling. Variable selection for semiparametric regression
models consists of two components: model selection for nonparametric components
and selection of significant variables for the parametric portion. Thus,
semiparametric variable selection is much more challenging than parametric
variable selection (e.g., linear and generalized linear models) because
traditional variable selection procedures including stepwise regression and the
best subset selection now require separate model selection for the
nonparametric components for each submodel. This leads to a very heavy
computational burden. In this paper, we propose a class of variable selection
procedures for semiparametric regression models using nonconcave penalized
likelihood. We establish the rate of convergence of the resulting estimate.
With proper choices of penalty functions and regularization parameters, we show
the asymptotic normality of the resulting estimate and further demonstrate that
the proposed procedures perform as well as an oracle procedure. A
semiparametric generalized likelihood ratio test is proposed to select
significant variables in the nonparametric component. We investigate the
asymptotic behavior of the proposed test and demonstrate that its limiting null
distribution follows a chi-square distribution which is independent of the
nuisance parameters. Extensive Monte Carlo simulation studies are conducted to
examine the finite sample performance of the proposed variable selection
procedures.
