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 -loss. Particularly, the effect of the dimensionality d of the state space on the nonparametric rate improves from to d compared to the case without singular value decay.
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.
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.
"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