Nonparametric estimation in hidden Markov models

This paper outlines a new procedure to perform nonparametric estimation in hidden Markov models. It is assumed that a Markov chain (Xk) is observed only through a process (Yk), where Yk is a noisy observation of f(Xk). We propose a maximum likelihood based procedure to estimate the function f using a block of observations. This paper shows the identifiability of the model under several assumptions on the Markov chain and on the function f. We also provide a proof of the consistency of the estimator of f as the number of observations grows to infinity. This consistency result relies on the Hellinger consistency of an estimator of the likelihood of the observations. Finally, we provide numerical experiments to highlight the performance of the estimator.