TY - JOUR
T1 - An oracle inequality for penalised projection estimation of Lévy densities from high-frequency observations
AU - Ueltzhöfer, Florian A.J.
AU - Klüppelberg, Claudia
N1 - Funding Information:
The authors wish to thank the referees and the associate editor for their detailed comments that helped to improve this paper significantly. It is also a pleasure to thank Jean Jacod for helpful discussions. The first author gratefully acknowledges support provided by the International Graduate School of Science and Engineering (IGSSE) of the Technische Universität München.
PY - 2011/12
Y1 - 2011/12
N2 - We consider a multivariate Lévy process given by the sum of a Brownian motion with drift and an independent time-homogeneous pure jump process governed by a Lévy density.We assume that observation of a sample path takes place on an equidistant discrete time grid. Following Grenander's method of sieves, we construct families of nonparametric projection estimators for the restriction of a Lévy density to bounded sets away from the origin. Moreover, we introduce a data-driven penalisation criterion to select an estimator within a given family, where we measure the estimation error in an L 2-norm. Furthermore, we give sufficient conditions on the penalty such that an oracle inequality holds. As an application, we prove adaptiveness for sufficiently smooth Lévy densities in some Sobolev space and explicitly derive the rate of convergence.
AB - We consider a multivariate Lévy process given by the sum of a Brownian motion with drift and an independent time-homogeneous pure jump process governed by a Lévy density.We assume that observation of a sample path takes place on an equidistant discrete time grid. Following Grenander's method of sieves, we construct families of nonparametric projection estimators for the restriction of a Lévy density to bounded sets away from the origin. Moreover, we introduce a data-driven penalisation criterion to select an estimator within a given family, where we measure the estimation error in an L 2-norm. Furthermore, we give sufficient conditions on the penalty such that an oracle inequality holds. As an application, we prove adaptiveness for sufficiently smooth Lévy densities in some Sobolev space and explicitly derive the rate of convergence.
KW - Adaptive model selection
KW - Lévy density
KW - Lévy process
KW - Nonparametric estimation
KW - Oracle inequality
UR - http://www.scopus.com/inward/record.url?scp=84855736704&partnerID=8YFLogxK
U2 - 10.1080/10485252.2011.581375
DO - 10.1080/10485252.2011.581375
M3 - Article
AN - SCOPUS:84855736704
SN - 1048-5252
VL - 23
SP - 967
EP - 989
JO - Journal of Nonparametric Statistics
JF - Journal of Nonparametric Statistics
IS - 4
ER -