Pivotal estimation via square-root Lasso in nonparametric regression
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Wang_Pivotal estimation.pdf
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Author(s)
Belloni, Alexandre
Wang, Lie
Chernozhukov, Victor V.
Date Issued
April 2014
Journal
Annals of Statistics
Publisher
Institute of Mathematical Statistics
Citation
Belloni, Alexandre, Victor Chernozhukov, and Lie Wang. “Pivotal Estimation via Square-Root Lasso in Nonparametric Regression.” Ann. Statist. 42, no. 2 (April 2014): 757–788.
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Author's final manuscript
Abstract
We propose a self-tuning √Lasso method that simultaneously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example, perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds for√Lasso including prediction norm rate and sparsity. Our analysis is based on new impact factors that are tailored for bounding prediction norm. In order to cover heteroscedastic non-Gaussian noise, we rely on moderate deviation theory for self-normalized sums to achieve Gaussian-like results under weak conditions. Moreover, we derive bounds on the performance of ordinary least square (ols) applied to the model selected by √Lasso accounting for possible misspecification of the selected model. Under mild conditions, the rate of convergence of ols post √Lasso is as good as √Lasso’s rate. As an application, we consider the use of √Lasso and ols post √Lasso as estimators of nuisance parameters in a generic semiparametric problem (nonlinear moment condition or Z-problem), resulting in a construction of √n-consistent and asymptotically normal estimators of the main parameters.
MIT Department
Massachusetts Institute of Technology. Department of Economics
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1214/14-AOS1204
