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UNIFORM ASYMPTOTICS AND CONFIDENCE REGIONS BASED ON THE ADAPTIVE LASSO WITH PARTIALLY CONSISTENT TUNING

Published online by Cambridge University Press:  08 April 2021

Nicolai Amann  and
Ulrike Schneider
Nicolai Amann
Affiliation:
University of Vienna
Ulrike Schneider*
Affiliation:
TU Wien
*
Address correspondence to Ulrike Schneider, Institute of Statistics and Mathematical Methods in Economics, TU Wien, Wiedner Hauptstraße 8/E105-2, A-1040 Vienna, Austria; e-mail: [email protected].
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Abstract

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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.

Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 6: SPECIAL ISSUE IN HONOR OF BENEDIKT M PÖTSCHER , December 2023 , pp. 1097 - 1122
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

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