Open Access
February 2016 Nonparametric finite translation hidden Markov models and extensions
Elisabeth Gassiat, Judith Rousseau
Bernoulli 22(1): 193-212 (February 2016). DOI: 10.3150/14-BEJ631


In this paper, we consider nonparametric finite translation hidden Markov models, or more generally finite translation mixtures with dependent latent variables. We prove that all the parameters of the model are identifiable as soon as the matrix that defines the joint distribution of two consecutive latent variables is non-singular and the translation parameters are distinct. Under this assumption, we provide a consistent estimator of the number of populations, of the translation parameters and of the distribution of two consecutive latent variables, which we prove to be asymptotically normally distributed under mild dependency assumptions. We propose a nonparametric estimator of the unknown translated density. In case the latent variables form a Markov chain, we prove that this estimator is minimax adaptive over regularity classes of densities.


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Elisabeth Gassiat. Judith Rousseau. "Nonparametric finite translation hidden Markov models and extensions." Bernoulli 22 (1) 193 - 212, February 2016.


Received: 1 June 2013; Revised: 1 December 2013; Published: February 2016
First available in Project Euclid: 30 September 2015

zbMATH: 06543267
MathSciNet: MR3449780
Digital Object Identifier: 10.3150/14-BEJ631

Keywords: dependent latent variable models , Hidden Markov models , nonparametric estimation , translation mixtures

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 1 • February 2016
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