Confidence Sets Based on the Lasso Estimator

In a linear regression model with fixed dimension, we construct confidence sets for the unknown parameter vector based on the Lasso estimator in finite samples as well as in an asymptotic setup, thereby quantifying estimation uncertainty of this estimator. In finite samples with Gaussian errors and asymptotically in the case where the Lasso estimator is tuned to perform conservative model-selection, we derive formulas for computing the minimal coverage probability over the entire parameter space for a large class of shapes for the confidence sets, thus enabling the construction of valid confidence sets based on the Lasso estimator in these settings. The choice of shape for the confidence sets and comparison with the confidence ellipse based on the least-squares estimator is also discussed. Moreover, in the case where the Lasso estimator is tuned to enable consistent model-selection, we give a simple confidence set with minimal coverage probability converging to one.