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45 results on '"Berlinet, Alain"'

1. Local linear regression for functional data

Author

Berlinet, Alain, Elamine, Abdallah, and Mas, André

Subjects
Mathematics - Statistics Theory
Abstract

We study a non linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method. We provide the asymptotic mean square error. Computations involve a linear inverse problem as well as a representation of the small ball probability of the data and are based on recent advances in this area. The rate of convergence of our estimate outperforms those already obtained in the literature on this model.

Published
2007

9. Theory

Author

Berlinet, Alain, Thomas-Agnan, Christine, Berlinet, Alain, and Thomas-Agnan, Christine

Published
2004
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20. About the asymptotic accuracy of Barron density estimates

Author

Berlinet, Alain, Vajda, Igor, and Meulen, Edward C. van der

Subjects
Information theory -- Analysis, Estimation theory -- Analysis
Abstract

By extending the information-theoretic arguments of previous papers dealing with the Barron-type density estimates, and their consistency in information divergence and chi-square divergence, the problem of consistency in Csiszar's [Phi]-divergence is motivated for general convex functions [Phi]. The problem of consistency in [Phi]-divergence is solved for all [Phi] with [Phi](0) < [infinity] and [Phi](t) = O(t ln t) when t [right arrow] [infinity]. The problem of consistency in the expected [Phi]-divergence is solved for all [Phi] with t [Phi](1/t) + [Phi](t) = O([t.sup.2]) when t [right arrow] [infinity]. Various stronger versions of these asymptotic restrictions are considered too. Assumptions about the model needed for the consistency are shown to depend on how strong these restrictions are. Index Terms - Barron density estimate, consistency in divergences and expected divergences, divergences of Csiszar, nonparametric density estimates, Renyi distances.

Published
1998

24. Régression linéaire locale pour variable fonctionnelle

Author

Berlinet, Alain, Elamine, Abdallah, Mas, André, Institut de Mathématiques et de Modélisation de Montpellier (I3M), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST]
Abstract

International audience; Dans ce travail, on s'intéresse à la régression non paramétrique locale pour des variables explicatives fonctionnelles. On propose tout d'abord un estimateur de l'opérateur de régression. La construction de cet estimateur est liée à la résolution d'un problème inverse linéaire. On établit des bornes de l'erreur quadratique moyenne en utilisant une méthode de décomposition classique. Cette EQM dépend de la fonction de petite boule de probabilité du régresseur au sujet de laquelle des hypothèses de type Gamma-variation sont posées.

Published
2009

25. FUNCTIONAL SUPERVISED CLASSIFICATION WITH WAVELETS

Author

Berlinet , Alain, Biau , Gérard, Rouviere , Laurent, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Laboratoire de Statistique Théorique et Appliquée (LSTA), Université Pierre et Marie Curie - Paris 6 (UPMC), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST-Université de Rennes 1 (UR1), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes 2 (UR2), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Statistique Théorique et Appliquée ( LSTA ), Université Pierre et Marie Curie - Paris 6 ( UPMC ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), and Gestionnaire, Hal Sorbonne Université

Subjects
[STAT.TH] Statistics [stat]/Statistics Theory [stat.TH], [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH], ComputingMilieux_MISCELLANEOUS
Abstract

International audience; Let X be a random variable taking values in a Hilbert space and let Y be a random label with values in {0, 1}. Given a collection of classification rules and a learning sample of independent copies of the pair (X, Y ), it is shown how to select optimally and consistently a classifier. As a general strategy, the learning sample observations are first expanded on a wavelet basis and the overall infinite dimension is reduced to a finite one via a suitable data-dependent thresholding. Then, a finite-dimensional classification rule is performed on the non-zero coefficients. Both the dimension and the classifier are automatically selected by data-splitting and empirical risk minimization. Applications of this technique to a signal discrimination problem involving speech recordings and simulated data are presented.

Published
2008

28. A Comparison Of Kernel Density Estimates

Author

Berlinet, Alain, Devroye, Luc, and HAL-SU, Gestionnaire

Subjects
Density estimation, convergence, smoothing factor, kernel estimate, nonparametric estimation, simulation, double kernel, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST]
Abstract

In the double kernel density estimate, the smoothing parameter h is chosen so as to minimize the L\ distance between two kernel density estimates having identical smoothing factors but different kernels. This method is known to be consistent for any density and to be asymptotically optimal for a certain smooth class of densities. We propose a plug-in modification of the estimate and introduce various other data-based bandwidth estimates. Finally, a simulation study is presented in which the new bandwidth selectors are compared with a host of well-known methods.

Published
1994

42. Cross-validated density estimates based on Kullback-Leibler information.

Author

Berlinet, Alain and Brunel, Elodie

Subjects
*STOCHASTIC convergence, *ESTIMATION theory, *INFORMATION theory, *BANDWIDTHS, *ESTIMATES, *STATISTICS
Abstract

The convergence of measure estimates in the sense of Kullback-Leibler divergence is required in many applications in decision and information theory. Recently, modified histograms have been shown to have good properties with respect to information divergences. For these estimates deterministic optimal bandwidths have been given, but no automatic smoothing procedure has been shown to be asymptotically optimal. In the present article, we consider the Kullback-Leibler cross-validation method for selecting the bin width of modified histograms. We analyze the behavior of the Kullback-Leibler divergence and of its expectation and prove that the cross-validated estimate is asymptotically optimal with respect to the Kullback-Leibler divergence. [ABSTRACT FROM AUTHOR]

Published
2004
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43. Iterated modified histograms as dynamical systems.

Author

Berlinet, Alain and Biau, Gérard

Subjects
*ESTIMATION theory, *MATHEMATICAL statistics, *STOCHASTIC processes, *DISTRIBUTION (Probability theory), *PROBABILITY theory, *SIMULATION methods & models
Abstract

A modified histogram estimate depending on a reference density has recently been proved to have good consistency properties according to several information theoretic criteria. In the present paper, this modified histogram is studied as a dynamical system in a functional space with the reference density as initial state. Properties of stationary densities of the system are investigated. It is proved, under appropriate assumptions, that the trajectories are almost surely stationary. Indeed, this result opens up the way for optimization of the estimate with respect to the reference density. Results of simulations are presented. [ABSTRACT FROM AUTHOR]

Published
2004
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45. Asymptotic Normality of L1-Error in Density Estimation

Author

Berlinet, Alain, Devroye, Luc, and Györfi, László

Abstract

Let fn be a histogram estimate constructed from a sample of i.i.d. real-valued random variables with common continuously differentiable density f. In this paper we prove a central limit theorem for the L1 error [image omitted] . We determine a positive constant 0 < σ2 ≤ 1 - 2/π in order that, under the usual conditions of consistency, the law of be asymptotically Gaussian , N(0, 1).

Published
1995
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