Sharp Oracle Inequalities for Aggregation of Affine Estimators
| Autor: | Arnak S. Dalalyan, Joseph Salmon |
|---|---|
| Přispěvatelé: | Centre de Recherche en Économie et Statistique (CREST), Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] (ENSAI)-École polytechnique (X)-École Nationale de la Statistique et de l'Administration Économique (ENSAE Paris)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2011 |
| Předmět: |
Statistics and Probability
Statistics::Theory model selection Heteroscedasticity Mathematics - Statistics Theory Statistics Theory (math.ST) 02 engineering and technology [STAT.OT]Statistics [stat]/Other Statistics [stat.ML] 01 natural sciences 010104 statistics & probability symbols.namesake 62G08 [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] minimax risk FOS: Mathematics 0202 electrical engineering electronic engineering information engineering oracle inequalities Applied mathematics 62G05 0101 mathematics 62G20 Mathematics 62C20 Model selection aggregation Estimator 020206 networking & telecommunications Regression analysis [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] exponential weights Minimax AMS 62G08 exponentially weighted aggregation Gaussian noise Adaptive estimator symbols regression Affine transformation Statistics Probability and Uncertainty |
| Zdroj: | Annals of Statistics Annals of Statistics, Institute of Mathematical Statistics, 2012, 40 (4), pp.2327-2355. ⟨10.1214/12-AOS1038⟩ Ann. Statist. 40, no. 4 (2012), 2327-2355 Annals of Statistics, 2012, 40 (4), pp.2327-2355. ⟨10.1214/12-AOS1038⟩ |
| ISSN: | 0090-5364 2168-8966 |
| DOI: | 10.1214/12-AOS1038⟩ |
| Popis: | International audience; 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. |
| Databáze: | OpenAIRE |
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