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We consider the adaptive Lasso estimator with componentwise tuning in theframework of a low-dimensional linear regression model. In our setting, atleast one of the components is penalized at the rate of consistent modelselection and certain components may not be penalized at all. We perform adetailed study of the consistency properties and the asymptotic distributionwhich includes the effects of componentwise tuning within a so-calledmoving-parameter framework. These results enable us to explicitly provide a set$\mathcal{M}$ such that every open superset acts as a confidence set withuniform asymptotic coverage equal to 1, whereas removing an arbitrarily smallopen set along the boundary yields a confidence set with uniform asymptoticcoverage equal to 0. The shape of the set $\mathcal{M}$ depends on theregressor matrix as well as the deviations within the componentwise tuningparameters. Our findings can be viewed as a broad generalization of P\"otscher& Schneider (2009, 2010) who considered distributional properties andconfidence intervals based on components of the adaptive Lasso estimator forthe case of orthogonal regressors.

Uniform Asymptotics and Confidence Regions Based on the Adaptive Lasso with Partially Consistent Tuning