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Robust parameter estimation for the Ornstein–Uhlenbeck process

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  • Sonja Rieder

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Abstract

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
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    References listed on IDEAS

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    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.
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    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.

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