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June 2013 Nonparametric inference on Lévy measures and copulas
Axel Bücher, Mathias Vetter
Ann. Statist. 41(3): 1485-1515 (June 2013). DOI: 10.1214/13-AOS1116

Abstract

In this paper nonparametric methods to assess the multivariate Lévy measure are introduced. Starting from high-frequency observations of a Lévy process $\mathbf{X} $, we construct estimators for its tail integrals and the Pareto–Lévy copula and prove weak convergence of these estimators in certain function spaces. Given $n$ observations of increments over intervals of length $\Delta_{n}$, the rate of convergence is $k_{n}^{-1/2}$ for $k_{n}=n\Delta_{n}$ which is natural concerning inference on the Lévy measure. Besides extensions to nonequidistant sampling schemes analytic properties of the Pareto–Lévy copula which, to the best of our knowledge, have not been mentioned before in the literature are provided as well. We conclude with a short simulation study on the performance of our estimators and apply them to real data.

Citation

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Axel Bücher. Mathias Vetter. "Nonparametric inference on Lévy measures and copulas." Ann. Statist. 41 (3) 1485 - 1515, June 2013. https://doi.org/10.1214/13-AOS1116

Information

Published: June 2013
First available in Project Euclid: 1 August 2013

zbMATH: 1273.62067
MathSciNet: MR3113819
Digital Object Identifier: 10.1214/13-AOS1116

Subjects:
Primary: 60F05 , 60G51 , 62H10
Secondary: 62G32 , 62M09

Keywords: copula , Lévy copula , Lévy measure , Lévy process , nonparametric statistics , Pareto–Lévy copula , weak convergence

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3 • June 2013
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