Confidence intervals that utilize sparsity

Abstract

We consider a linear regression model with orthogonal regressors and Gaussian random errors with known variance, in the low‐dimensional setting that the length of the regression parameter vector does not exceed the length of the response vector. We suppose that we have uncertain prior information that sparsity holds, that is, that many of the components of the regression parameter vector are zero. Our aim is to construct confidence intervals for the components of this vector with both the desired coverage probability and attractive expected length properties, particularly when sparsity holds. It is known that for fixed‐width confidence intervals centered on penalized maximum likelihood estimators, such as hard‐thresholding, LASSO and SCAD estimators, the achievement of the desired minimum coverage necessarily results in very unattractive expected length properties. We present confidence intervals, with data‐based widths, which achieve our aim. These confidence intervals dominate the usual confidence intervals with the same coverage probability, whenever the degree of sparsity exceeds a known value.