Adaptive estimation of the transition density of a particular hidden Markov chain

Abstract : We study the following model of hidden Markov chain: $Y_i=X_i+\varepsilon_i$, $i=1,\dots,n+1$ with $(X_i)$ a real-valued positive recurrent and stationary Markov chain and $(\varepsilon_i)_{1\leq i\leq n+1}$ a noise independent of the sequence $(X_i)$ having a known distribution. We present an adaptive estimator of the transition density based on the quotient of a deconvolution estimator of the density of $X_i$ and an estimator of the density of $(X_i,X_{i+1})$. These estimators are obtained by contrast minimization and model selection. We evaluate the $L2$ risk and its rate of convergence for ordinary smooth and supersmooth noise with regard to ordinary smooth and supersmooth chains. Some examples are also detailed.
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Journal of Multivariate Analysis, Elsevier, 2008, 99 (5), pp.787-814. <10.1016/j.jmva.2007.04.006>
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https://hal.archives-ouvertes.fr/hal-00115612
Contributeur : Claire Lacour <>
Soumis le : mercredi 22 novembre 2006 - 11:35:49
Dernière modification le : mardi 11 octobre 2016 - 12:01:26
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Claire Lacour. Adaptive estimation of the transition density of a particular hidden Markov chain. Journal of Multivariate Analysis, Elsevier, 2008, 99 (5), pp.787-814. <10.1016/j.jmva.2007.04.006>. <hal-00115612>

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