Least squares type estimation of the transition density of a particular hidden Markov chain
Claire Lacour · HAL (Le Centre pour la Communication Scientifique Directe) · 2008
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.