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

Free access article

Issue		ESAIM: PS Volume 6, 2002


Page(s)		211 - 238
DOI		10.1051/ps:2002012

ESAIM: P&S, November 2002, Vol. 6, pp. 211-238
DOI: 10.1051/ps:2002012

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

Fabienne Comte¹ and Florence Merlevède²

¹ Université Paris V, Laboratoire MAP5, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; comte@biomedicale.univ-paris5.fr.
² Université Paris VI, LSTA, 4 place Jussieu, 75252 Paris Cedex 05, France; merleve@ccr.jussieu.fr.

Abstract
In this paper, we study the problem of non parametric estimation of the stationary marginal density f of an $\alpha$ or a $\beta$ -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).

Mathematics Subject Classification. 62G07, 62M99.

Key words: Non parametric estimation, projection estimator, adaptive estimation, model selection, mixing processes, continuous time, discrete time.

© EDP Sciences, SMAI 2002

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