Nonparametric inference on Lévy measures and copulas

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.

Published in at http://dx.doi.org/10.1214/13-AOS1116 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Estimating the degree of activity of jumps in high frequency data
• Asymptotics of empirical copula processes under non-restrictive smoothness assumptions
• Empirical processes indexed by estimated functions
• Limit theorems for power variations of pure-jump processes with application to activity estimation
• Small-time expansions for the transition distributions of Lévy processes

Cited by in corpus (6)

• Nonparametric Bayesian inference for multidimensional compound Poisson processes
• Quantile estimation for Lévy measures
• Bootstrap confidence bands for spectral estimation of Lévy densities under high-frequency observations
• Series representations for bivariate time-changed L{é}vy models
• Low Frequency Lévy Copula Estimation
• Nonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations