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In this paper, we study the following model of hidden Markov chain:$Y_i=X_i+\epsilon_i$, $i=1,...,n+1$ with $(X_i)$ a real-valued stationaryMarkov chain and $(\epsilon_i)_{1\leq i\leq n+1}$ a noise having a knowndistribution and independent of the sequence $(X_i)$. We present an estimatorof the transition density obtained by minimization of an original contrast thattakes advantage of the regressive aspect of the problem. It is selected among acollection of projection estimators with a model selection method. The$L^2$-risk and its rate of convergence are evaluated for ordinary smooth noiseand some simulations illustrate the method. We obtain uniform risk bounds overclasses of Besov balls. In addition our estimation procedure requires no priorknowledge of the regularity of the true transition. Finally, our estimatorpermits to avoid the drawbacks of quotient estimators.

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