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Zbl 1171.62326
Zou, Hui
The adaptive lasso and its oracle properties. (English)
[J] J. Am. Stat. Assoc. 101, No. 476, 1418-1429 (2006). ISSN 0162-1459; ISSN 1537-274X/e

Summary: The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain scenarios where the lasso is inconsistent for variable selection. We then propose a new version of the lasso, called the adaptive lasso, where adaptive weights are used for penalizing different coefficients in the $\ell_1$ penalty. We show that the adaptive lasso enjoys the oracle properties; namely, it performs as well as if the true underlying model were given in advance. Similar to the lasso, the adaptive lasso is shown to be near-minimax optimal. Furthermore, the adaptive lasso can be solved by the same efficient algorithm for solving the lasso. We also discuss the extension of the adaptive lasso in generalized linear models and show that the oracle properties still hold under mild regularity conditions. As a byproduct of our theory, the nonnegative garotte is shown to be consistent for variable selection.

MSC 2000:: *62G08 Nonparametric regression
62G20 Nonparametric asymptotic efficiency
65C60 Computational problems in statistics
62G05 Nonparametric estimation
62J12 Generalized linear models

Keywords: asymptotic normality; lasso; minimax; oracle inequality; oracle procedure; variable selection

Cited in: Zbl 1168.62038 Zbl 1162.62063

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