Significance testing in non-sparse high-dimensional linear models

TL;DR

This paper introduces CorrT, a new inference method for high-dimensional linear models that remains reliable regardless of sparsity, heteroscedasticity, or model misspecification, outperforming existing methods.

Contribution

The paper proposes CorrT, a robust inferential method for high-dimensional linear models that works under various sparsity and heteroscedasticity conditions, unlike existing methods.

Findings

CorrT controls Type I error at the nominal level across models.

CorrT achieves near-zero Type II error in sparse and dense models.

Numerical experiments show CorrT outperforms state-of-the-art methods.

Abstract

In high-dimensional linear models, the sparsity assumption is typically made, stating that most of the parameters are equal to zero. Under the sparsity assumption, estimation and, recently, inference have been well studied. However, in practice, sparsity assumption is not checkable and more importantly is often violated; a large number of covariates might be expected to be associated with the response, indicating that possibly all, rather than just a few, parameters are non-zero. A natural example is a genome-wide gene expression profiling, where all genes are believed to affect a common disease marker. We show that existing inferential methods are sensitive to the sparsity assumption, and may, in turn, result in the severe lack of control of Type-I error. In this article, we propose a new inferential method, named CorrT, which is robust to model misspecification such as heteroscedasticity and lack of sparsity. CorrT is shown to have Type I error approaching the nominal level for \textit{any} models and Type II error approaching zero for sparse and many dense models. In fact, CorrT is also shown to be optimal in a variety of frameworks: sparse, non-sparse and hybrid models where sparse and dense signals are mixed. Numerical experiments show a favorable performance of the CorrT test compared to the state-of-the-art methods.