# Least squares type estimation of the transition density of a particular hidden Markov chain

Abstract : In this paper, 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 stationary Markov chain and $(\varepsilon_i)_{1\leq i\leq n+1}$ a noise having a known distribution and independent of the sequence $(X_i)$. We present an adaptive estimator of the transition density obtained by minimization of an original contrast taking advantage of the regressive aspect of the problem. It is selected among a collection of projection estimators with a model selection method. The $L^2$-risk and its rate of convergence are evaluated for ordinary smooth noise and some simulations illustrate the method. Our estimator allows to avoid the drawbacks of the quotient estimators.
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https://hal.archives-ouvertes.fr/hal-00180219
Contributor : Claire Lacour <>
Submitted on : Thursday, October 18, 2007 - 11:41:23 AM
Last modification on : Friday, September 20, 2019 - 4:34:03 PM
Long-term archiving on: Sunday, April 11, 2010 - 10:09:40 PM

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• HAL Id : hal-00180219, version 1

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Claire Lacour. Least squares type estimation of the transition density of a particular hidden Markov chain. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2008, 2, pp.1-39. ⟨hal-00180219⟩

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