# Confidence Intervals and Hypothesis Testing for High-Dimensional Regression

Adel Javanmard, Andrea Montanari.

Year: 2014, Volume: 15, Issue: 82, Pages: 2869−2909

#### 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 *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, made publicly available by BÃ¼hlmann et al. (2014).