## The Annals of Statistics

### Sharp oracle inequalities for aggregation of affine estimators

#### Abstract

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in nonparametric regression model with heteroscedastic Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads to sharp oracle inequalities in discrete but also in continuous settings. The framework is general enough to cover the combinations of various procedures such as least square regression, kernel ridge regression, shrinking estimators and many other estimators used in the literature on statistical inverse problems. As a consequence, we show that the proposed aggregate provides an adaptive estimator in the exact minimax sense without discretizing the range of tuning parameters or splitting the set of observations. We also illustrate numerically the good performance achieved by the exponentially weighted aggregate.

#### Article information

Source
Ann. Statist., Volume 40, Number 4 (2012), 2327-2355.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1358951384

Digital Object Identifier
doi:10.1214/12-AOS1038

Mathematical Reviews number (MathSciNet)
MR3059085

Zentralblatt MATH identifier
1257.62038

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62C20: Minimax procedures 62G05: Estimation 62G20: Asymptotic properties

#### Citation

Dalalyan, Arnak S.; Salmon, Joseph. Sharp oracle inequalities for aggregation of affine estimators. Ann. Statist. 40 (2012), no. 4, 2327--2355. doi:10.1214/12-AOS1038. https://projecteuclid.org/euclid.aos/1358951384

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