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| HAL: hal-00512304, version 2 |
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| Available versions: | v1 (2010-09-13) | v2 (2012-03-12) |
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| Optimal upper and lower bounds for the true and empirical excess risks in heteroscedastic least-squares regression |
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| Adrien Saumard 1, 2 |
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| (2010-08-29) |
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| We consider the estimation of a bounded regression function with nonparametric heteroscedastic noise. We are interested by the true and empirical excess risks of the least-squares estimator on a nite-dimensional vector space. For these quantities, we give upper and lower bounds in probability that are optimal at the rst order. Moreover, these bounds show the equivalence between the true and empirical excess risks when, among other things, the least-squares estimator is consistent in sup-norm towards the projection of the regression function onto the considered model. Consistency in sup-norm is then proved for suitable histogram models and more general models of piecewise polynomials that are endowed with a localized basis structure. |
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| 1: | Institut de Recherche Mathématique de Rennes (IRMAR) |
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CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne |
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| 2: | SELECT (INRIA Saclay - Ile de France) |
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INRIA – Université Paris XI - Paris Sud – CNRS : UMR |
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| Subject | : | Statistics/Other Statistics |
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| Least-squares regression – Heteroscedasticity – Excess risk – Lower bounds – Empirical process – M-estimation |
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| hal-00512304, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00512304 | |
| oai:hal.archives-ouvertes.fr:hal-00512304 | |
| From: Adrien Saumard | |
| Submitted on: Saturday, 10 March 2012 11:43:54 | |
| Updated on: Monday, 12 March 2012 08:46:07 | |
