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In this paper, we introduce a new estimator for the emission densities of anonparametric hidden Markov model. It is adaptive and minimax with respect toeach state's regularity--as opposed to globally minimax estimators, which adaptto the worst regularity among the emission densities. Our method is based onGoldenshluger and Lepski's methodology. It is computationally efficient andonly requires a family of preliminary estimators, without any restriction onthe type of estimators considered. We present two such estimators that allow toreach minimax rates up to a logarithmic term: a spectral estimator and a leastsquares estimator. We show how to calibrate it in practice and assess itsperformance on simulations and on real data.

State-by-state Minimax Adaptive Estimation for Nonparametric Hidden Markov Models