Confidence Intervals Using the Lasso

Abstract

Asymptotic linearity of a de-sparsified Lasso is established. This implies asymptotic normality under certain conditions and therefore can be used to construct confidence intervals for parameters of interest. Asymptotic linearity of groups of parameters, leading to confidence sets for groups, is also presented. Here, a the multivariate version of the square-root Lasso is invoked. The case of a linearized loss—applicable when the covariance matrix of the design is known—is briefly addressed as well. Throughout the chapter except for the last section, the design is considered as fixed.