The Annals of Statistics

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

Denis Belomestny

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In this article, the problem of semi-parametric inference on the parameters of a multidimensional Lévy process Lt 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 Lt, 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 Lt 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.

Article information

Ann. Statist., Volume 39, Number 4 (2011), 2205-2242.

First available in Project Euclid: 26 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F10: Point estimation
Secondary: 62J12: Generalized linear models 62F25: Tolerance and confidence regions 62H12: Estimation

Time-changed Lévy processes dependence pointwise and uniform rates of convergence composite function estimation


Belomestny, Denis. Statistical inference for time-changed Lévy processes via composite characteristic function estimation. Ann. Statist. 39 (2011), no. 4, 2205--2242. doi:10.1214/11-AOS901.

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