Electronic Journal of Statistics

The adaptive and the thresholded Lasso for potentially misspecified models (and a lower bound for the Lasso)

Sara van de Geer, Peter Bühlmann, and Shuheng Zhou

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We revisit the adaptive Lasso as well as the thresholded Lasso with refitting, in a high-dimensional linear model, and study prediction error, q-error (q{1,2}), and number of false positive selections. Our theoretical results for the two methods are, at a rather fine scale, comparable. The differences only show up in terms of the (minimal) restricted and sparse eigenvalues, favoring thresholding over the adaptive Lasso. As regards prediction and estimation, the difference is virtually negligible, but our bound for the number of false positives is larger for the adaptive Lasso than for thresholding. We also study the adaptive Lasso under beta-min conditions, which are conditions on the size of the coefficients. We show that for exact variable selection, the adaptive Lasso generally needs more severe beta-min conditions than thresholding. Both the two-stage methods add value to the one-stage Lasso in the sense that, under appropriate restricted and sparse eigenvalue conditions, they have similar prediction and estimation error as the one-stage Lasso but substantially less false positives. Regarding the latter, we provide a lower bound for the Lasso with respect to false positive selections.

Article information

Electron. J. Statist., Volume 5 (2011), 688-749.

First available in Project Euclid: 25 July 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J07: Ridge regression; shrinkage estimators
Secondary: 62G08: Nonparametric regression

Adaptive Lasso estimation prediction restricted eigenvalue thresholding variable selection


van de Geer, Sara; Bühlmann, Peter; Zhou, Shuheng. The adaptive and the thresholded Lasso for potentially misspecified models (and a lower bound for the Lasso). Electron. J. Statist. 5 (2011), 688--749. doi:10.1214/11-EJS624. https://projecteuclid.org/euclid.ejs/1311600467

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