A Likelihood Ratio Framework for High Dimensional Semiparametric Regression

Abstract

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.

51 pages, 1 figure, 2 tables

References in corpus (15)

• Nearly unbiased variable selection under minimax concave penalty
• Simultaneous analysis of Lasso and Dantzig selector
• Challenges of Big Data Analysis
• On the conditions used to prove oracle results for the Lasso
• Consistency of the group Lasso and multiple kernel learning
• Sure independence screening in generalized linear models with NP-dimensionality
• High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity
• Support union recovery in high-dimensional multivariate regression
• Strong oracle optimality of folded concave penalized estimation
• Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection
• Missing values: sparse inverse covariance estimation and an extension to sparse regression
• Exact Post-Selection Inference for Sequential Regression Procedures
• Inference in High Dimensions with the Penalized Score Test
• Post-Selection Inference for Generalized Linear Models with Many Controls
• Group Lasso for generalized linear models in high dimension

Cited by in corpus (5)

• Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach
• Sparse Nonlinear Regression: Parameter Estimation and Asymptotic Inference
• Bayes factor consistency
• On Fast Convergence of Proximal Algorithms for SQRT-Lasso Optimization: Don't Worry About Its Nonsmooth Loss Function
• On Quadratic Convergence of DC Proximal Newton Algorithm for Nonconvex Sparse Learning in High Dimensions