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


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”.


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


Download Citation

Chiara Amorino. "Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes." Electron. J. Statist. 15 (2) 5067 - 5116, 2021.


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

Digital Object Identifier: 10.1214/21-EJS1913

Primary: 62G07 , 62G20
Secondary: 60J74

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

Vol.15 • No. 2 • 2021
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