Minimax adaptive estimation of nonparametric hidden Markov models

Abstract

We consider stationary hidden Markov models with finite state space and nonparametric modeling of the emission distributions. It has remained unknown until very recently that such models are identifiable. In this paper, we propose a new penalized least-squares esti-mator for the emission distributions which is statistically optimal and practically tractable. We prove a non asymptotic oracle inequality for our nonparametric estimator of the emission distributions. A consequence is that this new estimator is rate minimax adaptive up to a logarithmic term. Our methodology is based on projections of the emission distributions onto nested subspaces of increasing complexity. The popular spectral estimators are unable to achieve the optimal rate but may be used as initial points in our procedure. Simulations are given that show the improvement obtained when applying the least-squares minimization consecutively to the spectral estimation.

The nonparametric spectral method is independently studied in the arXiv preprint arXiv:1507.06510

References in corpus (3)

• Identifiability of parameters in latent structure models with many observed variables
• Finite state space non parametric Hidden Markov Models are in general identifiable
• Nonparametric identification and maximum likelihood estimation for hidden Markov model

Cited by in corpus (4)

• Multiple Testing in Nonparametric Hidden Markov Models: An Empirical Bayes Approach
• Fundamental limits for learning hidden Markov model parameters
• Identifiability and consistent estimation of nonparametric translation hidden Markov models with general state space
• Non Parametric Hidden Markov Models with Finite State Space: Posterior Concentration Rates