## The Annals of Statistics

### Pivotal estimation via square-root Lasso in nonparametric regression

#### Abstract

We propose a self-tuning $\sqrt{\mathrm {Lasso}}$ 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 $\sqrt{\mathrm {Lasso}}$ 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 $\sqrt{\mathrm {Lasso}}$ accounting for possible misspecification of the selected model. Under mild conditions, the rate of convergence of ols post $\sqrt{\mathrm {Lasso}}$ is as good as $\sqrt{\mathrm {Lasso}}$’s rate. As an application, we consider the use of $\sqrt{\mathrm {Lasso}}$ and ols post $\sqrt{\mathrm {Lasso}}$ as estimators of nuisance parameters in a generic semiparametric problem (nonlinear moment condition or $Z$-problem), resulting in a construction of $\sqrt{n}$-consistent and asymptotically normal estimators of the main parameters.

#### Article information

Source
Ann. Statist., Volume 42, Number 2 (2014), 757-788.

Dates
First available in Project Euclid: 20 May 2014

https://projecteuclid.org/euclid.aos/1400592177

Digital Object Identifier
doi:10.1214/14-AOS1204

Mathematical Reviews number (MathSciNet)
MR3210986

Zentralblatt MATH identifier
1321.62030

Subjects
Primary: 62G05: Estimation 62G08: Nonparametric regression
Secondary: 62G35: Robustness

#### Citation

Belloni, Alexandre; Chernozhukov, Victor; Wang, Lie. Pivotal estimation via square-root Lasso in nonparametric regression. Ann. Statist. 42 (2014), no. 2, 757--788. doi:10.1214/14-AOS1204. https://projecteuclid.org/euclid.aos/1400592177

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#### Supplemental materials

• Supplementary material: Supplementary material. The material contains deferred proofs, additional theoretical results on convergence rates in $\ell_{2},\ell_{1}$ and $\ell_{\infty}$, lower bound on the prediction rate, and Monte-Carlo simulations.