Partial Penalized Empirical Likelihood Ratio Test Under Sparse Case

A consistent test via the partial penalized empirical likelihood approach for the parametric hypothesis testing under the sparse case, called the partial penalized empirical likelihood ratio（PPELR） test, is proposed in this paper. Our results are demonstrated for the mean vector in multivariate analysis and regression coefficients in linear models, respectively. And we establish its asymptotic distributions under the null hypothesis and the local alternatives of order n-1/2 under regularity conditions. Meanwhile, the oracle property of the partial penalized empirical likelihood estimator also holds. The proposed PPELR test statistic performs as well as the ordinary empirical likelihood ratio test statistic and outperforms the full penalized empirical likelihood ratio test statistic in term of size and power when the null parameter is zero. Moreover, the proposed method obtains the variable selection 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 method in hypothesis testing and variable selection.