Novel Inference Procedures for Non-Standard High-Dimensional Regression Models

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Statistical theory of hypothesis testing plays a fundamental role in virtually all scientific studies. In the era of big data, high-dimensional data are ubiquitous in many scientific fields such as natural sciences, social sciences, medicine, and public health. Therefore, modern applications often involve hypothesis testing under high dimensions, which calls for new statistical inference theory. As regression is the most popular statistical analysis tool in applications, some recent work has been focused on hypothesis testing in high dimensional least squares regression. However, it is well-known that the standard least squares regression model has severe limitations in real applications. This research aims to develop new statistical inference theory for more flexible high dimensional regression models. This research focuses on the development of inference theory for several important non-standard regression models under ultra-high dimensions. Specifically, the PI will develop tests for testing linear hypotheses under three models: high dimensional expectile regression, high dimensional heteroscedastic regression, and robust high dimensional regression. Asymptotic distributions of the test statistics will be established rigorously. The theoretical study will fill important gaps in the high-dimensional statistics literature. A unified efficient algorithm will be developed to tackle the computational challenges. The research will provide principled tools for studying expectile functions, for examining the heterogeneity in high dimensional data and for performing robust inference. Research training opportunities for graduate students will be provided.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.