Significance testing in non-sparse high-dimensional linear models
This addresses the issue of unreliable inference in high-dimensional statistics for researchers and practitioners when sparsity assumptions are not met, such as in genomics, and is incremental by improving robustness over existing methods.
The paper tackles the problem of significance testing in high-dimensional linear models where the sparsity assumption is often violated, proposing a new method called CorrT that achieves Type I error control at the nominal level for any model and reduces Type II error to zero for sparse and dense models, with numerical experiments showing favorable performance compared to state-of-the-art methods.
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.