# Significance testing in non-sparse high-dimensional linear models

arXiv: https://arxiv.org/abs/1610.02122 · HTML: https://mlsanity.com/paper/1610.02122/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2016.

## Built on by (most cited first, 3 of 3)

- [Surrogate Assisted Semi-supervised Inference for High Dimensional Risk Prediction](https://mlsanity.com/paper/2105.01264/) (2021)
- [Comments on Leo Breiman's paper 'Statistical Modeling: The Two Cultures' (Statistical Science, 2001, 16(3), 199-231)](https://mlsanity.com/paper/2103.11327/) (2021)
- [Classification of high-dimensional data with spiked covariance matrix structure](https://mlsanity.com/paper/2110.01950/) (2021)

## Abstract (arXiv)

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 textitany 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.