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+@?"i,i=1,...,n+1 with (X"i) a real-valued positive recurrent and stationary Markov chain, and (@?"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.

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