Conditioning to stay positive; local time; Lévy processes; occupation time; optimal estimation; self-similarity; supremum; weak limit theorems; Statistics and Probability; Statistics, Probability and Uncertainty
Abstract :
[en] In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a Lévy process. More specifically, we investigate the asymptotic theory for the conditional mean and conditional median estimators of the supremum/infimum of a linear Brownian motion and a strictly stable Lévy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators studied in e.g. [6, 14, 29, 30, 38]. Furthermore, we discuss pre-estimation of the parameters of the underly-ing models, which is required for practical implementation of the proposed statistics.
Disciplines :
Mathematics
Author, co-author :
Ivanovs, Jevgenijs; Department of Mathematics, Aarhus University, Denmark
PODOLSKIJ, Mark ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
External co-authors :
yes
Language :
English
Title :
Optimal estimation of the supremum and occupation times of a self-similar Lévy process∗
Publication date :
2022
Journal title :
Electronic Journal of Statistics
eISSN :
1935-7524
Publisher :
Institute of Mathematical Statistics
Volume :
16
Issue :
1
Pages :
892 - 934
Peer reviewed :
Peer Reviewed verified by ORBi
European Projects :
H2020 - 815703 - STAMFORD - Statistical Methods For High Dimensional Diffusions
Name of the research project :
Statistical Methods For High Dimensional Diffusions
Funders :
ERC - European Research Council Union Européenne
Funding number :
815703
Funding text :
∗The financial support of ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High Dimensional Diffusions” and Sapere Aude Starting Grant 8049-00021B “Distributional Robustness in Assessment of Extreme Risk” is gratefully acknowledged.
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