A likelihood ratio framework for high-dimensional semiparametric regression

We propose a new inferential framework for high-dimensional semiparametric generalized linear models. This framework addresses a variety of challenging problems in high-dimensional data analysis, including incomplete data, selection bias and heterogeneity. Our work has three main contributions: (i) We develop a regularized statistical chromatography approach to infer the parameter of interest under the proposed semiparametric generalized linear model without the need of estimating the unknown base measure function. (ii) We propose a new likelihood ratio based framework to construct post-regularization confidence regions and tests for the low dimensional components of high-dimensional parameters. Unlike existing post-regularization inferential methods, our approach is based on a novel directional likelihood. (iii) We develop new concentration inequalities and normal approximation results for U-statistics with unbounded kernels, which are of independent interest. We further extend the theoretical results to the problems of missing data and multiple datasets inference. Extensive simulation studies and real data analysis are provided to illustrate the proposed approach.