Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes

We study the problem of the non-parametric estimation for the density - of the stationary distribution of the multivariate stochastic differential equation with jumps (X), when the dimension d is such that d - 3. From the continuous observation of the sampling path on [0,T]we show that, under anisot...

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Detalles Bibliográficos
Autor: Amorino, Chiara
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2021
País:España
Institución:Universitat Pompeu Fabra
Repositorio:Repositorio Digital de la UPF
OAI Identifier:oai:repositori.upf.edu:10230/71772
Acceso en línea:http://hdl.handle.net/10230/71772
http://dx.doi.org/10.1214/21-EJS1913
Access Level:acceso abierto
Palabra clave:Convergence rate
Density estimation
Ergodic diffusion with jumps
Lévy driven SDE
Non-parametric statistics
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spelling Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classesAmorino, ChiaraConvergence rateDensity estimationErgodic diffusion with jumpsLévy driven SDENon-parametric statisticsWe study the problem of the non-parametric estimation for the density - of the stationary distribution of the multivariate stochastic differential equation with jumps (X), when the dimension d is such that d - 3. 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.The author gratefully acknowledges financial support of ERC Consolidator Grant 815703 "STAMFORD: Statistical Methods for High Dimensional Diffusions".Institute of Mathematical Statistics2025202520212025info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/10230/71772http://dx.doi.org/10.1214/21-EJS1913reponame:Repositorio Digital de la UPFinstname:Universitat Pompeu FabraInglésElectronic Journal of Statistics. 2021;15(2):5067-5116info:eu-repo/grantAgreement/EC/H2020/815703Creative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:repositori.upf.edu:10230/717722026-06-12T07:21:37Z
dc.title.none.fl_str_mv Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
title Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
spellingShingle Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
Amorino, Chiara
Convergence rate
Density estimation
Ergodic diffusion with jumps
Lévy driven SDE
Non-parametric statistics
title_short Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
title_full Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
title_fullStr Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
title_full_unstemmed Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
title_sort Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
dc.creator.none.fl_str_mv Amorino, Chiara
author Amorino, Chiara
author_facet Amorino, Chiara
author_role author
dc.subject.none.fl_str_mv Convergence rate
Density estimation
Ergodic diffusion with jumps
Lévy driven SDE
Non-parametric statistics
topic Convergence rate
Density estimation
Ergodic diffusion with jumps
Lévy driven SDE
Non-parametric statistics
description We study the problem of the non-parametric estimation for the density - of the stationary distribution of the multivariate stochastic differential equation with jumps (X), when the dimension d is such that d - 3. 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.
publishDate 2021
dc.date.none.fl_str_mv 2021
2025
2025
2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10230/71772
http://dx.doi.org/10.1214/21-EJS1913
url http://hdl.handle.net/10230/71772
http://dx.doi.org/10.1214/21-EJS1913
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Electronic Journal of Statistics. 2021;15(2):5067-5116
info:eu-repo/grantAgreement/EC/H2020/815703
dc.rights.none.fl_str_mv Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Institute of Mathematical Statistics
publisher.none.fl_str_mv Institute of Mathematical Statistics
dc.source.none.fl_str_mv reponame:Repositorio Digital de la UPF
instname:Universitat Pompeu Fabra
instname_str Universitat Pompeu Fabra
reponame_str Repositorio Digital de la UPF
collection Repositorio Digital de la UPF
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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