Adaptive estimation of the stationary density of discrete and continuous time mixing processes

Fabienne Comte; Florence Merlevède

ESAIM: Probability and Statistics (2010)

  • Volume: 6, page 211-238
  • ISSN: 1292-8100

Abstract

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In this paper, we study the problem of non parametric estimation of the stationary marginal density f of an α or a β-mixing process, observed either in continuous time or in discrete time. We present an unified framework allowing to deal with many different cases. We consider a collection of finite dimensional linear regular spaces. We estimate f using a projection estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized contrast. We state non asymptotic risk bounds, regarding to the integrated quadratic risk, for our estimators, in both cases of mixing. We show that they are adaptive in the minimax sense over a large class of Besov balls. In discrete time, we also provide a result for model selection among an exponentially large collection of models (non regular case).

How to cite

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Comte, Fabienne, and Merlevède, Florence. "Adaptive estimation of the stationary density of discrete and continuous time mixing processes." ESAIM: Probability and Statistics 6 (2010): 211-238. <http://eudml.org/doc/104289>.

@article{Comte2010,
abstract = { In this paper, we study the problem of non parametric estimation of the stationary marginal density f of an α or a β-mixing process, observed either in continuous time or in discrete time. We present an unified framework allowing to deal with many different cases. We consider a collection of finite dimensional linear regular spaces. We estimate f using a projection estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized contrast. We state non asymptotic risk bounds, regarding to the integrated quadratic risk, for our estimators, in both cases of mixing. We show that they are adaptive in the minimax sense over a large class of Besov balls. In discrete time, we also provide a result for model selection among an exponentially large collection of models (non regular case). },
author = {Comte, Fabienne, Merlevède, Florence},
journal = {ESAIM: Probability and Statistics},
keywords = {Non parametric estimation; projection estimator; adaptive estimation; model selection; mixing processes; continuous time; discrete time.},
language = {eng},
month = {3},
pages = {211-238},
publisher = {EDP Sciences},
title = {Adaptive estimation of the stationary density of discrete and continuous time mixing processes},
url = {http://eudml.org/doc/104289},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Comte, Fabienne
AU - Merlevède, Florence
TI - Adaptive estimation of the stationary density of discrete and continuous time mixing processes
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 211
EP - 238
AB - In this paper, we study the problem of non parametric estimation of the stationary marginal density f of an α or a β-mixing process, observed either in continuous time or in discrete time. We present an unified framework allowing to deal with many different cases. We consider a collection of finite dimensional linear regular spaces. We estimate f using a projection estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized contrast. We state non asymptotic risk bounds, regarding to the integrated quadratic risk, for our estimators, in both cases of mixing. We show that they are adaptive in the minimax sense over a large class of Besov balls. In discrete time, we also provide a result for model selection among an exponentially large collection of models (non regular case).
LA - eng
KW - Non parametric estimation; projection estimator; adaptive estimation; model selection; mixing processes; continuous time; discrete time.
UR - http://eudml.org/doc/104289
ER -

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