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February 2019 Consistent order estimation for nonparametric hidden Markov models
Luc Lehéricy
Bernoulli 25(1): 464-498 (February 2019). DOI: 10.3150/17-BEJ993


We consider the problem of estimating the number of hidden states (the order) of a nonparametric hidden Markov model (HMM). We propose two different methods and prove their almost sure consistency without any prior assumption, be it on the order or on the emission distributions. This is the first time a consistency result is proved in such a general setting without using restrictive assumptions such as a priori upper bounds on the order or parametric restrictions on the emission distributions. Our main method relies on the minimization of a penalized least squares criterion. In addition to the consistency of the order estimation, we also prove that this method yields rate minimax adaptive estimators of the parameters of the HMM – up to a logarithmic factor. Our second method relies on estimating the rank of a matrix obtained from the distribution of two consecutive observations. Finally, numerical experiments are used to compare both methods and study their ability to select the right order in several situations.


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Luc Lehéricy. "Consistent order estimation for nonparametric hidden Markov models." Bernoulli 25 (1) 464 - 498, February 2019.


Received: 1 April 2017; Revised: 1 September 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007214
MathSciNet: MR3892326
Digital Object Identifier: 10.3150/17-BEJ993

Keywords: Hidden Markov model , least squares method , Model selection , Nonparametric density estimation , order estimation , Spectral method

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability


Vol.25 • No. 1 • February 2019
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