  We consider estimation and inference in a single index regression model with
an unknown but smooth link function. In contrast to the standard approach of
using kernels or regression splines, we use smoothing splines to estimate the
smooth link function. We develop a method to compute the penalized least
squares estimators (PLSEs) of the parametric and the nonparametric components
given independent and identically distributed (i.i.d.)~data. We prove the
consistency and find the rates of convergence of the estimators. We establish
asymptotic normality under under mild assumption and prove asymptotic
efficiency of the parametric component under homoscedastic errors. A finite
sample simulation corroborates our asymptotic theory. We also analyze a car
mileage data set and a Ozone concentration data set. The identifiability and
existence of the PLSEs are also investigated.
