Open Access
October 2005 Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes
G. Jongbloed, F.H. Van Der Meulen, A.W. Van Der Vaart
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Bernoulli 11(5): 759-791 (October 2005). DOI: 10.3150/bj/1130077593


We consider nonparametric estimation of the Lévy measure of a hidden Lévy process driving a stationary Ornstein-Uhlenbeck process which is observed at discrete time points. This Lévy measure can be expressed in terms of the canonical function of the stationary distribution of the Ornstein-Uhlenbeck process, which is known to be self-decomposable. We propose an estimator for this canonical function based on a preliminary estimator of the characteristic function of the stationary distribution. We provide a support-reduction algorithm for the numerical computation of the estimator, and show that the estimator is asymptotically consistent under various sampling schemes. We also define a simple consistent estimator of the intensity parameter of the process. Along the way, a nonparametric procedure for estimating a self-decomposable density function is constructed, and it is shown that the Ornstein-Uhlenbeck process is β-mixing. Some general results on uniform convergence of random characteristic functions are included.


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G. Jongbloed. F.H. Van Der Meulen. A.W. Van Der Vaart. "Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes." Bernoulli 11 (5) 759 - 791, October 2005.


Published: October 2005
First available in Project Euclid: 23 October 2005

zbMATH: 1084.62080
MathSciNet: MR2172840
Digital Object Identifier: 10.3150/bj/1130077593

Keywords: Lévy process , self-decomposability , support-reduction algorithm , uniform convergence of characteristic functions

Rights: Copyright © 2005 Bernoulli Society for Mathematical Statistics and Probability

Vol.11 • No. 5 • October 2005
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