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

We study the following model of hidden Markov chain: with (Xi) a real-valued positive recurrent and stationary Markov chain, and (ɛi)1⩽i⩽n+1 a noise independent of the sequence (Xi) 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 Xi and an estimator of the density of (Xi,Xi+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.