Confidence Sets Based on Penalized Maximum Likelihood Estimators in Gaussian Regression

TL;DR

This paper analyzes confidence intervals derived from penalized maximum likelihood estimators like LASSO and adaptive LASSO in Gaussian regression, focusing on coverage properties, interval lengths, and asymptotic behavior.

Contribution

It provides finite-sample coverage analysis and compares interval lengths for various penalized estimators, introducing a simple asymptotic confidence interval for the sparse case.

Findings

Symmetric intervals are shortest for known variance.

Interval length order: hard-thresholding > adaptive LASSO > LASSO > standard MLE.

Intervals based on sparse estimators are significantly larger, by an order of magnitude.

Abstract

Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the maximum likelihood estimator. In the case where the penalized estimators are tuned to possess the `sparsity property', the intervals based on these estimators are larger than the standard interval by an order of magnitude. Furthermore, a simple asymptotic confidence interval construction in the `sparse' case, that also applies to the smoothly clipped absolute deviation estimator, is discussed. The results for the known-variance case are shown to carry over to the unknown-variance case in an appropriate asymptotic sense.