  This paper is concerned with estimation and inference for ultrahigh
dimensional partially linear single-index models. The presence of high
dimensional nuisance parameter and nuisance unknown function makes the
estimation and inference problem very challenging. In this paper, we first
propose a profile partial penalized least squares estimator and establish the
sparsity, consistency and asymptotic representation of the proposed estimator
in ultrahigh dimensional setting. We then propose an $F$-type test statistic
for parameters of primary interest and show that the limiting null distribution
of the test statistic is $\chi^2$ distribution, and the test statistic can
detect local alternatives, which converge to the null hypothesis at the
root-$n$ rate. We further propose a new test for the specification testing
problem of the nonparametric function. The test statistic is shown to be
asymptotically normal. Simulation studies are conducted to examine the finite
sample performance of the proposed estimators and tests. A real data example is
used to illustrate the proposed procedures.
