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Sharp Oracle Inequalities for Aggregation of Affine Estimators

Abstract : We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric 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 neither discretizing the range of tuning parameters nor splitting the set of observations. We also illustrate numerically the good performance achieved by the exponentially weighted aggregate.
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Contributor : Arnak Dalalyan Connect in order to contact the contributor
Submitted on : Monday, February 27, 2012 - 10:47:14 PM
Last modification on : Saturday, June 19, 2021 - 3:33:31 AM
Long-term archiving on: : Monday, May 28, 2012 - 3:05:36 AM


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Arnak S. Dalalyan, Joseph Salmon. Sharp Oracle Inequalities for Aggregation of Affine Estimators. Annals of Statistics, Institute of Mathematical Statistics, 2012, 40 (4), pp.2327-2355. ⟨10.1214/12-AOS1038⟩. ⟨hal-00587225v3⟩



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