Nonparametric estimation of a trend based upon sampled continuous processes

David Degras

doi:10.1016/j.crma.2008.12.016

Statistics

Nonparametric estimation of a trend based upon sampled continuous processes
[Estimation non paramétrique d'une tendance à partir de réalisations d'un processus à temps continu]

Présenté par : Paul Deheuvels

David Degras ¹

¹ Laboratoire de statistique théorique et appliquée, Université Paris 6, boîte 158, 175, rue du Chevaleret, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 191-194.

Résumé
Abstract

Soit X= ${X (t), t \in [0, T]}$ un processus aléatoire du second ordre dont on observe n réalisations indépendantes sur une grille de p points déterministes. Sous de faibles conditions de régularité sur les trajectoires de X, nous prouvons la normalité asymptotique d'estimateurs non paramétriques de la tendance $μ = E X$ dans l'espace $C ([0, T])$ lorsque $n, p \to \infty$ , puis nous obtenons des bandes de confiance simultanées approchées pour μ à l'aide de la théorie des processus Gaussiens.

Let $X = {X (t), t \in [0, T]}$ be a second order random process of which n independent realizations are observed on a fixed grid of p time points. Under mild regularity assumptions on the sample paths of X, we show the asymptotic normality of suitable nonparametric estimators of the trend function $μ = E X$ in the space $C ([0, T])$ as $n, p \to \infty$ and, using Gaussian process theory, we derive approximate simultaneous confidence bands for μ.

Reçu le : 2008-07-19
Accepté le : 2008-12-19
Publié le : 2009-01-29

DOI : 10.1016/j.crma.2008.12.016

Affiliations des auteurs :

David Degras ¹

¹ Laboratoire de statistique théorique et appliquée, Université Paris 6, boîte 158, 175, rue du Chevaleret, 75013 Paris, France

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David Degras. Nonparametric estimation of a trend based upon sampled continuous processes. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 191-194. doi : 10.1016/j.crma.2008.12.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.12.016/

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David Degras Simultaneous confidence bands for the mean of functional data, Wiley Interdisciplinary Reviews. WIREs Computational Statistics, Volume 9 (2017) no. 3, p. 15 (Id/No e1397) | DOI:10.1002/wics.1397 | Zbl:1545.62030
Hervé Cardot; David Degras; Etienne Josserand Confidence bands for Horvitz-Thompson estimators using sampled noisy functional data, Bernoulli, Volume 19 (2013) no. 5A, pp. 2067-2097 | DOI:10.3150/12-bej443 | Zbl:1457.62394
H. Cardot; E. Josserand Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling, Biometrika, Volume 98 (2011) no. 1, p. 107 | DOI:10.1093/biomet/asq070
Florentina Bunea; Andrada E. Ivanescu; Marten H. Wegkamp Adaptive inference for the mean of a Gaussian process in functional data, Journal of the Royal Statistical Society. Series B. Statistical Methodology, Volume 73 (2011) no. 4, pp. 531-558 | DOI:10.1111/j.1467-9868.2010.00768.x | Zbl:1226.62073

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