We consider the adaptive Lasso estimator with componentwise tuning in the framework of a low-dimensional linear regression model. In our setting, at least one of the components is penalized at the rate of consistent model selection and certain components may not be penalized at all. We perform a detailed study of the consistency properties and the asymptotic distribution which 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 removing an arbitrarily small open set along the boundary yields 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 broad generalization of Pötscher and Schneider (2009, Journal of Statistical Planning and Inference 139, 2775–2790; 2010, Electronic Journal of Statistics 4, 334–360), who considered distributional properties and confidence intervals based on components of the adaptive Lasso estimator for the case of orthogonal regressors.

我们考虑在低维线性回归模型框架中进行组件调整的自适应 Lasso 估计器。在我们的设置中，至少有一个组件会以一致模型选择的速率受到惩罚，而某些组件可能根本不会受到惩罚。我们对一致性属性和渐进分布进行了详细研究，其中包括所谓的移动参数框架内的组件调整的影响。这些结果使我们能够明确地提供一个集合$\mathcal {M}$，使得每个开超集都充当均匀渐近覆盖等于 1 的置信集，而沿边界删除任意小的开集会产生一个均匀渐进覆盖的置信集渐近覆盖等于 0。集合$\mathcal {M}$的形状取决于回归矩阵以及分量调整参数内的偏差。我们的研究结果可以被视为 Pötscher 和 Schneider (2009, Journal of Statistical Planning and Inference 139, 2775–2790; 2010, Electronic Journal of Statistics 4, 334–360) 的广泛概括，他们考虑了基于正交回归量情况下的自适应 Lasso 估计器的组成部分。