Bernoulli

  • Bernoulli
  • Volume 11, Number 5 (2005), 759-791.

Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes

G. Jongbloed, F.H. Van Der Meulen, and A.W. Van Der Vaart

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Abstract

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.

Article information

Source
Bernoulli, Volume 11, Number 5 (2005), 759-791.

Dates
First available in Project Euclid: 23 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.bj/1130077593

Digital Object Identifier
doi:10.3150/bj/1130077593

Mathematical Reviews number (MathSciNet)
MR2172840

Zentralblatt MATH identifier
1084.62080

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

Citation

Jongbloed, G.; Van Der Meulen, F.H.; Van Der Vaart, A.W. Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli 11 (2005), no. 5, 759--791. doi:10.3150/bj/1130077593. https://projecteuclid.org/euclid.bj/1130077593


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