We propose a likelihood ratio based 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 heterogeneous multitask learning. 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 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.
In particular, the framework naturally handles generic regularized estimators
with nonconvex penalty functions and it can be used to infer least false
parameters under misspecified models. (iii) We develop new concentration
inequalities and normal approximation results for U-statistics with unbounded
kernels, which are of independent interest. We demonstrate the consequences of
the general theory by using an example of missing data problem. Extensive
simulation studies and real data analysis are provided to illustrate our
proposed approach.

提出一种基于似然比的高维半参数广义线性模型的推理框架，包括使用规则化统计制备模型、构建后规则化的置信区间和测试、发展新的浓缩不等式和正态逼近结果，以及应用于缺失数据问题的示例。