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Robust parameter estimation for the Ornstein–Uhlenbeck process

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  • Sonja Rieder


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    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

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    Article provided by Springer & Società Italiana di Statistica in its journal Statistical Methods & Applications.

    Volume (Year): 21 (2012)
    Issue (Month): 4 (November)
    Pages: 411-436

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    Handle: RePEc:spr:stmapp:v:21:y:2012:i:4:p:411-436
    DOI: 10.1007/s10260-012-0195-2
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    1. 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.
    2. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Croux, C. & Dehon, C., 2010. "Influence Functions of the Spearman and Kendall Correlation Measures," Discussion Paper 2010-40, Tilburg University, Center for Economic Research.
    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.
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