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2021 Nonasymptotic control of the MLE for misspecified nonparametric hidden Markov models
Luc Lehéricy
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Electron. J. Statist. 15(2): 4916-4965 (2021). DOI: 10.1214/21-EJS1890

Abstract

Finite state space hidden Markov models are flexible tools to model phenomena with complex time dependencies: any process distribution can be approximated by a hidden Markov model with enough hidden states. We consider the problem of estimating an unknown process distribution using nonparametric hidden Markov models in the misspecified setting, that is when the data-generating process may not be a hidden Markov model. We show that when the true distribution is exponentially mixing and satisfies a forgetting assumption, the maximum likelihood estimator recovers the best approximation of the true distribution. We prove a finite sample bound on the resulting error and show that it is optimal in the minimax sense–up to logarithmic factors–when the model is well specified.

Acknowledgments

I am grateful to Élisabeth Gassiat for her precious advice and insightful discussions. I would also like to thank the anonymous referee for his patience and very helpful review.

Citation

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Luc Lehéricy. "Nonasymptotic control of the MLE for misspecified nonparametric hidden Markov models." Electron. J. Statist. 15 (2) 4916 - 4965, 2021. https://doi.org/10.1214/21-EJS1890

Information

Received: 1 January 2021; Published: 2021
First available in Project Euclid: 6 December 2021

Digital Object Identifier: 10.1214/21-EJS1890

Subjects:
Primary: 62M09
Secondary: 62G20 , 62G35

Keywords: Hidden Markov model , maximum likelihood estimator , misspecified model , Model selection , nonparametric statistics , Oracle inequality

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Vol.15 • No. 2 • 2021
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