# Estimation non paramétrique adaptative pour les chaînes de Markov et les chaînes de Markov cachées

Abstract : In this thesis, we consider a Markov chain $(X_i)$ with continuous state space which is assumed positive recurrent and stationary. The aim is to estimate the transition density $\Pi$ defined by $\Pi (X, y) dy=P(X_{i+1}\in dy|X_i=x)$. We use model selection to construct adaptive estimators. We work in the minimax framework on $L^2$ and we are interested in the rates of convergence obtained when transition density is supposed to be regular. The integrated risk of our estimators is bounded thanks to control of empirical processes by a concentration inequality of Talagrand. In a first part, we suppose that the chain is directly observed. Two different estimators are introduced, one by quotient, the other minimizing a least squares contrast and also taking into account the anisotropy of the problem. In a second part, we treat the case of noisy observations $Y_1, \dots, Y_{n+1}$ where $Y_i=X_i+ \varepsilon_i$ with $(\varepsilon_i)$ a noise independent of the chain $(X_i)$. We generalize to this case the two previous estimators. Some simulations illustrate the performances of the estimators.
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Cited literature [89 references]

https://tel.archives-ouvertes.fr/tel-00180107
Contributor : Claire Lacour <>
Submitted on : Wednesday, October 17, 2007 - 4:41:53 PM
Last modification on : Friday, September 20, 2019 - 4:34:03 PM
Long-term archiving on: Sunday, April 11, 2010 - 10:01:57 PM

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• HAL Id : tel-00180107, version 1

### Citation

Claire Lacour. Estimation non paramétrique adaptative pour les chaînes de Markov et les chaînes de Markov cachées. Mathématiques [math]. Université René Descartes - Paris V, 2007. Français. ⟨tel-00180107⟩

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