Confidence Intervals and Hypothesis Testing for High-Dimensional Regression

TL;DR

This paper introduces a new method for constructing confidence intervals and p-values in high-dimensional linear regression, overcoming the challenge of unknown parameter estimate distributions without assuming special design matrix structures.

Contribution

It proposes a de-biased estimator approach that yields nearly optimal confidence intervals and test power without restrictive design matrix assumptions.

Findings

Confidence intervals are nearly optimal in size.

The method achieves nearly optimal power for hypothesis testing.

Validated on synthetic and genomic data.

Abstract

Fitting high-dimensional statistical models often requires the use of non-linear parameter estimation procedures. As a consequence, it is generally impossible to obtain an exact characterization of the probability distribution of the parameter estimates. This in turn implies that it is extremely challenging to quantify the \emph{uncertainty} associated with a certain parameter estimate. Concretely, no commonly accepted procedure exists for computing classical measures of uncertainty and statistical significance as confidence intervals or $p$-values for these models. We consider here high-dimensional linear regression problem, and propose an efficient algorithm for constructing confidence intervals and $p$-values. The resulting confidence intervals have nearly optimal size. When testing for the null hypothesis that a certain parameter is vanishing, our method has nearly optimal power. Our approach is based on constructing a `de-biased' version of regularized M-estimators. The new construction improves over recent work in the field in that it does not assume a special structure on the design matrix. We test our method on synthetic data and a high-throughput genomic data set about riboflavin production rate.