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Parametric Estimation for Subordinators and Induced OU Processes

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  • GEURT JONGBLOED
  • FRANK H. VAN DER MEULEN

Abstract

. Consider a stationary sequence of random variables with infinitely divisible marginal law, characterized by its Lévy density. We analyse the behaviour of a so‐called cumulant M‐estimator, in case this Lévy density is characterized by a Euclidean (finite dimensional) parameter. Under mild conditions, we prove consistency and asymptotic normality of the estimator. The estimator is considered in the situation where the data are increments of a subordinator as well as the situation where the data consist of a discretely sampled Ornstein–Uhlenbeck (OU) process induced by the subordinator. We illustrate our results for the Gamma‐process and the Inverse‐Gaussian OU process. For these processes we also explain how the estimator can be computed numerically.

Suggested Citation

  • Geurt Jongbloed & Frank H. Van Der Meulen, 2006. "Parametric Estimation for Subordinators and Induced OU Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(4), pages 825-847, December.
    • Handle: RePEc:bla:scjsta:v:33:y:2006:i:4:p:825-847
    DOI: 10.1111/j.1467-9469.2006.00498.x
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    Cited by:

    1. Taufer, Emanuele & Leonenko, Nikolai & Bee, Marco, 2011. "Characteristic function estimation of Ornstein-Uhlenbeck-based stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 55(8), pages 2525-2539, August.
    2. Kevin W. Lu, 2022. "Calibration for multivariate Lévy-driven Ornstein-Uhlenbeck processes with applications to weak subordination," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 365-396, July.
    3. Mélina Bec & Claire Lacour, 2015. "Adaptive pointwise estimation for pure jump Lévy processes," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 229-256, October.
    4. Shu, Yin & Feng, Qianmei & Liu, Hao, 2019. "Using degradation-with-jump measures to estimate life characteristics of lithium-ion battery," Reliability Engineering and System Safety, Elsevier, vol. 191(C).
    5. Yanfeng Wu & Jianqiang Hu & Xiangyu Yang, 2022. "Moment estimators for parameters of Lévy‐driven Ornstein–Uhlenbeck processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(4), pages 610-639, July.
    6. Emanuele Taufer, 2008. "Characteristic function estimation of non-Gaussian Ornstein-Uhlenbeck processes," DISA Working Papers 0805, Department of Computer and Management Sciences, University of Trento, Italy, revised 07 Jul 2008.
    7. Shibin Zhang & Xinsheng Zhang, 2013. "A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 89-103, February.
    8. Hiroki Masuda, 2009. "Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(1), pages 181-195, March.
    9. Comte, F. & Genon-Catalot, V., 2009. "Nonparametric estimation for pure jump Lévy processes based on high frequency data," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4088-4123, December.

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