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 P\"otscher & Schneider (2009, 2010) who considered distributional properties and confidence intervals based on components of the adaptive Lasso estimator for the case of orthogonal regressors.

翻译：我们考虑在低维线性回归模型框架内进行组件调整的适应性激光测算仪。 在我们的设置中, 至少有一个组件受到一致模式选择率的处罚, 某些组件可能不会受到任何处罚。 我们对一致性属性和无症状分布进行详细研究, 其中包括在所谓的移动参数框架内进行组件调整的效果。 这些结果可以让我们明确提供一套 $\ mathcal{ M}$, 这样每张打开的超集都作为信任设置, 其统一的无线回归覆盖等于 1, 而删除边界沿线的任意小开放集则产生一个信任配置, 统一的无线覆盖率等于 0。 设定的元集的形状取决于递增矩阵以及组件调整参数中的偏差。 我们的发现可以被视为P\ “ otscher & Schneider ” (2009, 2010) 的广泛概括性, 他们根据适应性激光测量仪的组件考虑分布属性和信任度间隔。