Statistical inference for time-changed Lévy processes via composite characteristic function estimation

Abstract

In this article, the problem of semi-parametric inference on the parameters of a multidimensional Lévy process $L_t$ with independent components based on the low-frequency observations of the corresponding time-changed Lévy process $L_{\mathcal{T}(t)}$, where $\mathcal{T}$ is a nonnegative, nondecreasing real-valued process independent of $L_t$, is studied. We show that this problem is closely related to the problem of composite function estimation that has recently gotten much attention in statistical literature. Under suitable identifiability conditions, we propose a consistent estimate for the Lévy density of $L_t$ and derive the uniform as well as the pointwise convergence rates of the estimate proposed. Moreover, we prove that the rates obtained are optimal in a minimax sense over suitable classes of time-changed Lévy models. Finally, we present a simulation study showing the performance of our estimation algorithm in the case of time-changed Normal Inverse Gaussian (NIG) Lévy processes.

Published in at http://dx.doi.org/10.1214/11-AOS901 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Multivariate Archimedean copulas, $d$-monotone functions and ${\ell }_1$-norm symmetric distributions
• Rate-optimal estimation for a general class of nonparametric regression models with unknown link functions
• Spectral estimation of the fractional order of a Lévy process
• Nonparametric estimation of composite functions
• Asymptotic results and statistical procedures for time-changed Lévy processes sampled at hitting times

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