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Adaptive estimation of the stationary densityof discrete and continuous time mixingprocesses

Published online by Cambridge University Press:  15 November 2002

Fabienne Comte  and
Florence Merlevède
Fabienne Comte
Affiliation:
Université Paris V, Laboratoire MAP5, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; [email protected].
Florence Merlevède
Affiliation:
Université Paris VI, LSTA, 4 place Jussieu, 75252 Paris Cedex 05, France; [email protected].
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Abstract

In this paper, we study the problem of non parametric estimationof the stationary marginal density f of an α or aβ-mixing process, observed either in continuous time or indiscrete time. We present an unified framework allowing to dealwith many different cases. We consider a collection of finitedimensional linear regular spaces. We estimate f using aprojection estimator built on a data driven selected linear spaceamong the collection. This data driven choice is performed via theminimization of a penalized contrast. We state non asymptotic riskbounds, regarding to the integrated quadratic risk, for ourestimators, in both cases of mixing. We show that they areadaptive in the minimax sense over a large class of Besov balls.In discrete time, we also provide a result for model selectionamong an exponentially large collection of models (non regularcase).

Keywords

Non parametric estimationprojection estimatoradaptive estimationmodel selectionmixing processescontinuoustimediscrete time.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 6: New directions in Time Series Analysis (Guest Editor: Philippe Soulier)New directions in Time Series Analysis (Guest Editor: Philippe Soulier) , 2002 , pp. 211 - 238
Copyright
© EDP Sciences, SMAI, 2002

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