  This work is concern with testing the low-dimensional parameters of interest
with divergent dimensional data and variable selection for the rest under the
sparse case. A consistent test via the partial penalized likelihood approach,
called the partial penalized likelihood ratio test statistic is derived, and
its asymptotic distributions under the null hypothesis and the local
alternatives of order $n^{-1/2}$ are obtained under some regularity conditions.
Meanwhile, the oracle property of the partial penalized likelihood estimator
also holds. The proposed partial penalized likelihood ratio test statistic
outperforms the full penalized likelihood ratio test statistic in term of size
and power, and performs as well as the classical likelihood ratio test
statistic. Moreover, the proposed method obtains the variable selection results
as well as the p-values of testing. Numerical simulations and an analysis of
Prostate Cancer data confirm our theoretical findings and demonstrate the
promising performance of the proposed partial penalized likelihood in
hypothesis testing and variable selection.
