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2021 Spectral thresholding for the estimation of Markov chain transition operators
Matthias Löffler, Antoine Picard
Electron. J. Statist. 15(2): 6281-6310 (2021). DOI: 10.1214/21-EJS1935

Abstract

We consider nonparametric estimation of the transition operator P of a Markov chain and its transition density p where the singular values of P are assumed to decay exponentially fast. This is for instance the case for periodised, reversible multi-dimensional diffusion processes observed in low frequency.

We investigate the performance of a spectral hard thresholded Galerkin-type estimator for P and p, discarding most of the estimated singular triplets. The construction is based on smooth basis functions such as wavelets or B-splines. We show its statistical optimality by establishing matching minimax upper and lower bounds in L2-loss. Particularly, the effect of the dimensionality d of the state space on the nonparametric rate improves from 2d to d compared to the case without singular value decay.

Funding Statement

M. Löffler was supported by ERC grant UQMSI/647812 and EPSRC grant EP/L016516/1. Further partial support by ETH Foundations of Data Science is gratefully acknowledged. A. Picard would like to thank the Statslab and ERC grant UQMSI/647812 for supporting him during the undertaking of parts of this work while visiting R. Nickl’s research group from February to June 2018.

Acknowledgments

Both authors are grateful to R. Nickl, K. Abraham and S. Wang for helpful discussions, to Y. Sun for pointing out a mistake in a previous version of the paper and to two anonymous referees for their helpful comments, suggestions and remarks.

Citation

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Matthias Löffler. Antoine Picard. "Spectral thresholding for the estimation of Markov chain transition operators." Electron. J. Statist. 15 (2) 6281 - 6310, 2021. https://doi.org/10.1214/21-EJS1935

Information

Received: 1 October 2020; Published: 2021
First available in Project Euclid: 27 December 2021

Digital Object Identifier: 10.1214/21-EJS1935

Subjects:
Primary: 62G05
Secondary: 62C20

Keywords: low rank , Markov chain , minimax rates of convergence , nonparametric estimation , SDE , Transition density , transition operator

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Vol.15 • No. 2 • 2021
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