Article (Scientific journals)
Optimal convergence rates for the invariant density estimation of jump-diffusion processes
AMORINO, Chiara; Nualart, Eulalia
2022In ESAIM: Probability and Statistics
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Keywords :
Minimax risk; Convergence rate; non-parametric statistics; ergodic diffusion with jumps; Levy driven SDE; invariant density estimation
Abstract :
[en] We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and d = 2. We consider a class of jump diffusion processes whose invariant density belongs to some Hölder space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate 1/T, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for d = 1 and is equal to log T/T for d = 2. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates { 1/T , log T/T } in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.
Disciplines :
Mathematics
Author, co-author :
AMORINO, Chiara ;  University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Nualart, Eulalia
External co-authors :
yes
Language :
English
Title :
Optimal convergence rates for the invariant density estimation of jump-diffusion processes
Publication date :
January 2022
Journal title :
ESAIM: Probability and Statistics
ISSN :
1292-8100
eISSN :
1262-3318
Publisher :
EDP Sciences, France
Peer reviewed :
Peer Reviewed verified by ORBi
Funders :
ERC Consolidator Grant 815703 STAMFORD: Statistical Methods for High Dimensional Diffusions, MINECO grant PGC2018-101643-B-I00 and Ayudas Fundacion BBVA a Equipos de Investigación Científica 2017
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