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We consider the estimation of a bounded regression function with nonparametric heteroscedastic noise and random design. We study the true and empirical excess risks of the least-squares estimator on finite-dimensional vector spaces. We give upper and lower bounds on these quantities that are nonasymptotic and optimal to first order, allowing the dimension to depend on sample size. 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 with the projection of the regression function onto the considered model. Consistency in the 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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Autor(es)

Saumard, Adrien - 

Id.: 55207165

Idioma: inglés  - 

Versión: 1.0

Estado: Final

Tipo:  application/pdf - 

Palabras claveLeast -  squares regression - 

Tipo de recurso: Text  - 

Tipo de Interactividad: Expositivo

Nivel de Interactividad: muy bajo

Audiencia: Estudiante  -  Profesor  -  Autor  - 

Estructura: Atomic

Coste: no

Copyright: sí

: Copyright 2012 Institute of Mathematical Statistics

Formatos:  application/pdf - 

Requerimientos técnicos:  Browser: Any - 

Relación: [References] 1935-7524

Fecha de contribución: 26-ene-2013

Contacto:

Localización:
* Electron. J. Statist. 6 (2012), 579-655
* doi:10.1214/12-EJS679

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  1. Efficiency of the V-fold model selection for localized bases Many interesting functional bases, such as piecewise polynomials or wavelets, are examples of locali...
  2. Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases We investigate the optimality for model selection of the so-calledslope heuristics, $V$-fold cross-v...
  3. Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases We investigate the optimality for model selection of the so-called slope heuristics, V-fold cross-va...
  4. On optimality of empirical risk minimization in linear aggregation 37 pages
  5. On optimality of empirical risk minimization in linear aggregation 37 pages

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