The Annals of Statistics

Convergence rates of least squares regression estimators with heavy-tailed errors

Qiyang Han and Jon A. Wellner

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We study the performance of the least squares estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a $p$th moment ($p\geq1$). In such a heavy-tailed regression setting, we show that if the model satisfies a standard “entropy condition” with exponent $\alpha\in(0,2)$, then the $L_{2}$ loss of the LSE converges at a rate

\[\mathcal{O}_{\mathbf{P}}\bigl(n^{-\frac{1}{2+\alpha}}\vee n^{-\frac{1}{2}+\frac{1}{2p}}\bigr).\] Such a rate cannot be improved under the entropy condition alone.

This rate quantifies both some positive and negative aspects of the LSE in a heavy-tailed regression setting. On the positive side, as long as the errors have $p\geq1+2/\alpha$ moments, the $L_{2}$ loss of the LSE converges at the same rate as if the errors are Gaussian. On the negative side, if $p<1+2/\alpha$, there are (many) hard models at any entropy level $\alpha$ for which the $L_{2}$ loss of the LSE converges at a strictly slower rate than other robust estimators.

The validity of the above rate relies crucially on the independence of the covariates and the errors. In fact, the $L_{2}$ loss of the LSE can converge arbitrarily slowly when the independence fails.

The key technical ingredient is a new multiplier inequality that gives sharp bounds for the “multiplier empirical process” associated with the LSE. We further give an application to the sparse linear regression model with heavy-tailed covariates and errors to demonstrate the scope of this new inequality.

Article information

Ann. Statist., Volume 47, Number 4 (2019), 2286-2319.

Received: February 2018
Revised: May 2018
First available in Project Euclid: 21 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62G05: Estimation

Multiplier empirical process multiplier inequality nonparametric regression least squares estimation sparse linear regression heavy-tailed errors


Han, Qiyang; Wellner, Jon A. Convergence rates of least squares regression estimators with heavy-tailed errors. Ann. Statist. 47 (2019), no. 4, 2286--2319. doi:10.1214/18-AOS1748.

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Supplemental materials

  • Supplement: Additional proofs. In the supplement [26], we provide detailed proofs for (i) Theorem 4, (ii) the impossibility results Propositions 1 and 3 and (iii) all remaining lemmas.