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

We study the following model of hidden Markov chain: Y i = X i + ɛ i , i = 1 , … , n + 1 with ( X i ) a real-valued positive recurrent and stationary Markov chain, and ( ɛ i ) 1 ⩽ i ⩽ 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 L 2 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.