IDEAS home Printed from
   My bibliography  Save this article

Robust parameter estimation for the Ornstein–Uhlenbeck process


  • Sonja Rieder



In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein–Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008 ). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981 ). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et al. (Stat Methods Appl 19:333–354, 2010 ), Rieder (Robust asymptotic statistics. Springer, New York, 1994 ), Rieder ( 2003 ), and Staab ( 1984 ). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969 ) and compare it by means of simulations with MLE and M-estimator. Copyright Springer-Verlag 2012

Suggested Citation

  • Sonja Rieder, 2012. "Robust parameter estimation for the Ornstein–Uhlenbeck process," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(4), pages 411-436, November.
  • Handle: RePEc:spr:stmapp:v:21:y:2012:i:4:p:411-436
    DOI: 10.1007/s10260-012-0195-2

    Download full text from publisher

    File URL:
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    1. Swensen, Anders Rygh, 1985. "The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend," Journal of Multivariate Analysis, Elsevier, vol. 16(1), pages 54-70, February.
    2. Marc Hallin & Christophe Koell & Bas Werker, 2000. "Optimal inference for discretely observed semiparametric Ornstein-Uhlenbeck processes," ULB Institutional Repository 2013/2097, ULB -- Universite Libre de Bruxelles.
    3. Mancini, Loriano & Ronchetti, Elvezio & Trojani, Fabio, 2005. "Optimal Conditionally Unbiased Bounded-Influence Inference in Dynamic Location and Scale Models," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 628-641, June.
    4. Matthias Kohl & Peter Ruckdeschel & Helmut Rieder, 2010. "Infinitesimally Robust estimation in general smoothly parametrized models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 19(3), pages 333-354, August.
    5. Croux, C. & Dehon, C., 2010. "Influence Functions of the Spearman and Kendall Correlation Measures," Discussion Paper 2010-40, Tilburg University, Center for Economic Research.
    6. Tadeusz Bednarski, 2010. "Fréchet differentiability in statistical inference for time series," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 19(4), pages 517-528, November.
    7. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    8. Yuji Sakamoto & Nakahiro Yoshida, 1998. "Asymptotic Expansion of M ‐Estimator Over Wiener Space," Statistical Inference for Stochastic Processes, Springer, vol. 1(1), pages 85-103, January.
    9. Christophe Croux & Catherine Dehon, 2010. "Influence functions of the Spearman and Kendall correlation measures," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 19(4), pages 497-515, November.
    10. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(04), pages 627-627, November.
    11. Nakahiro Yoshida, 1990. "Asymptotic behavior of M-estimator and related random field for diffusion process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 221-251, June.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Marco Riani & Andrea Cerioli & Francesca Torti, 2014. "On consistency factors and efficiency of robust S-estimators," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 356-387, June.
    2. David Stibůrek, 2016. "Hypotheses testing about the drift parameter in linear stochastic differential equation driven by stable processes," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 25(3), pages 433-452, August.


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:stmapp:v:21:y:2012:i:4:p:411-436. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla) or (Rebekah McClure). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.