Asymptotic Confidence Regions Based on the Adaptive Lasso with Partial Consistent Tuning

Abstract

We construct confidence sets based on an adaptive Lasso estimator with componentwise tuning in the framework of a low-dimensional linear regression model. We consider the case where at least one of the components is penalized at the rate of consistent model selection and where certain components may not be penalized at all. We perform a detailed study of the consistency properties and the asymptotic distribution that includes the effects of componentwise tuning within a so-called moving-parameter framework. These results enable us to explicitly provide a set \(\mathcal{M}\) such that every open superset acts as a confidence set with uniform asymptotic coverage equal to 1 whereas every proper closed subset with non-empty interior is a confidence set with uniform asymptotic coverage equal to 0. The shape of the set \(\mathcal{M}\) depends on the regressor matrix as well as the deviations within the componentwise tuning parameters. Our findings can be viewed as a generalization of P\"otscher & Schneider (2010) who considered confidence intervals based on components of the adaptive Lasso estimator for the case of orthogonal regressors.