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2021 Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes
Chiara Amorino
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Electron. J. Statist. 15(2): 5067-5116 (2021). DOI: 10.1214/21-EJS1913

Abstract

We study the problem of the non-parametric estimation for the density π of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt)0tT, when the dimension d is such that d3. From the continuous observation of the sampling path on [0,T] we show that, under anisotropic Hölder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular, they are as fast as the ones found by Dalalyan and Reiss [11] for the estimation of the invariant density in the case without jumps under isotropic Hölder smoothness constraints. Moreover, they are faster than the ones found by Strauch [32] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Hölder smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.

Funding Statement

The author gratefully acknowledges financial support of ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High Dimensional Diffusions”.

Acknowledgments

The author is very grateful to Arnaud Gloter who supported the project and helped to improve the paper.

Citation

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Chiara Amorino. "Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes." Electron. J. Statist. 15 (2) 5067 - 5116, 2021. https://doi.org/10.1214/21-EJS1913

Information

Received: 1 December 2020; Published: 2021
First available in Project Euclid: 6 December 2021

Digital Object Identifier: 10.1214/21-EJS1913

Subjects:
Primary: 62G07 , 62G20
Secondary: 60J74

Keywords: convergence rate , Density estimation , ergodic diffusion with jumps , Lévy driven SDE , Non-parametric statistics

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Vol.15 • No. 2 • 2021
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