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

This paper considers the problem of estimating the emission densities of a nonparametric finite state space hidden Markov model in a way that is state-by-state adaptive and leads to minimax rates for each emission density--as opposed to globally minimax estimators, which adapt to the worst regularity among the emission densities. We propose a model selection procedure based on the Goldenschluger-Lepski method. Our method is computationally efficient and only requires a family of preliminary estimators, without any restriction on the type of estimators considered. We present two such estimators that allow to reach minimax rates up to a logarithmic term: a spectral estimator and a least squares estimator. Finally, numerical experiments assess the performance of the method and illustrate how to calibrate it in practice. Our method is not specific to hidden Markov models and can be applied to nonparametric multiview mixture models.