Institute of Mathematical Statistics Lecture Notes - Monograph Series

Nonparametric Estimation for Lévy Models Based on Discrete-Sampling

José E. Figueroa-López

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Abstract

A Lévy model combines a Brownian motion with drift and a pure-jump homogeneous process such as a compound Poisson process. The estimation of the Lévy density, the infinite-dimensional parameter controlling the jump dynamics of the process, is studied under a discrete-sampling scheme. In that case, the jumps are latent variables whose statistical properties can in principle be assessed when the frequency of observations increase to infinity. We propose nonparametric estimators for the Lévy density following Grenander’s method of sieves. The associated problem of selecting a suitable approximating sieve is subsequently investigated using regular piece-wise polynomials as sieves and assuming standard smoothness conditions on the Lévy density. By sampling the process at a high enough frequency relative to the time horizon T, we show that it is feasible to choose the dimension of the sieve so that the rate of convergence of the risk of estimation off the origin is the best possible from a minimax point of view, and even if the estimation were based on the whole sample path of the process. The sampling frequency necessary to attain the optimal minimax rate is explicitly identified. The proposed method is illustrated by simulation experiments in the case of variance Gamma processes.

Chapter information

Source
Javier Rojo, ed., Optimality: The Third Erich L. Lehmann Symposium (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2009), 117-146

Dates
First available in Project Euclid: 3 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.lnms/1249305327

Digital Object Identifier
doi:10.1214/09-LNMS5709

Zentralblatt MATH identifier
1271.62067

Subjects
Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Keywords
Lévy processes nonparametric estimation model selection

Rights
Copyright © 2009, Institute of Mathematical Statistics

Citation

Rojo, Javier. Nonparametric Estimation for Lévy Models Based on Discrete-Sampling. Optimality, 117--146, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2009. doi:10.1214/09-LNMS5709. https://projecteuclid.org/euclid.lnms/1249305327


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