Partial Penalized Likelihood Ratio Test under Sparse Case

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.