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Parameter Estimation for a Bidimensional Partially Observed Ornstein–Uhlenbeck Process with Biological Application

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  • BENJAMIN FAVETTO
  • ADELINE SAMSON

Abstract

. We consider a bidimensional Ornstein–Uhlenbeck process to describe the tissue microvascularization in anti‐cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is to estimate the SDE parameters. We use the main advantage of a one‐dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion, which allows to implement an easy numerical maximization of the likelihood. Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the maximum likelihood estimator. We show that this ARMA property can be generalized to a higher dimensional underlying Ornstein–Uhlenbeck diffusion. We compare this estimator with the one obtained by the well‐known expectation maximization algorithm on simulated data. Our estimation methods can be directly applied to other biological contexts such as drug pharmacokinetics or hormone secretions.

Suggested Citation

  • Benjamin Favetto & Adeline Samson, 2010. "Parameter Estimation for a Bidimensional Partially Observed Ornstein–Uhlenbeck Process with Biological Application," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(2), pages 200-220, June.
    • Handle: RePEc:bla:scjsta:v:37:y:2010:i:2:p:200-220
    DOI: 10.1111/j.1467-9469.2009.00679.x
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    References listed on IDEAS

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    1. Valentine Genon‐Catalot & Thierry Jeantheau & Catherine Laredo, 2003. "Conditional Likelihood Estimators for Hidden Markov Models and Stochastic Volatility Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(2), pages 297-316, June.
    2. Genon-Catalot, Valentine & Laredo, Catherine, 2006. "Leroux's method for general hidden Markov models," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 222-243, February.
    3. Ruiz, Esther, 1994. "Quasi-maximum likelihood estimation of stochastic volatility models," Journal of Econometrics, Elsevier, vol. 63(1), pages 289-306, July.
    4. R. H. Shumway & D. S. Stoffer, 1982. "An Approach To Time Series Smoothing And Forecasting Using The Em Algorithm," Journal of Time Series Analysis, Wiley Blackwell, vol. 3(4), pages 253-264, July.
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    Cited by:

    1. Peilun He & Gareth W. Peters & Nino Kordzakhia & Pavel V. Shevchenko, 2024. "Multi-Factor Function-on-Function Regression of Bond Yields on WTI Commodity Futures Term Structure Dynamics," Papers 2412.05889, arXiv.org.
    2. Nina Munkholt Jakobsen & Michael Sørensen, 2015. "Efficient Estimation for Diffusions Sampled at High Frequency Over a Fixed Time Interval," CREATES Research Papers 2015-33, Department of Economics and Business Economics, Aarhus University.
    3. Becheri, I.G., 2012. "Limiting experiments for panel-data and jump-diffusion models," Other publications TiSEM 7e53f6cf-fab1-4f86-9e5d-b, Tilburg University, School of Economics and Management.
    4. Picchini, Umberto & Ditlevsen, Susanne, 2011. "Practical estimation of high dimensional stochastic differential mixed-effects models," Computational Statistics & Data Analysis, Elsevier, vol. 55(3), pages 1426-1444, March.
    5. Narci, Romain & Delattre, Maud & Larédo, Catherine & Vergu, Elisabeta, 2021. "Inference for partially observed epidemic dynamics guided by Kalman filtering techniques," Computational Statistics & Data Analysis, Elsevier, vol. 164(C).
    6. Oscar García, 2019. "Estimating reducible stochastic differential equations by conversion to a least-squares problem," Computational Statistics, Springer, vol. 34(1), pages 23-46, March.
    7. Peilun He & Nino Kordzakhia & Gareth W. Peters & Pavel V. Shevchenko, 2024. "Multi-Factor Polynomial Diffusion Models and Inter-Temporal Futures Dynamics," Papers 2409.19386, arXiv.org.
    8. Samson, Adeline & Thieullen, Michèle, 2012. "A contrast estimator for completely or partially observed hypoelliptic diffusion," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2521-2552.
    9. Karol Binkowski & Peilun He & Nino Kordzakhia & Pavel Shevchenko, 2021. "On the Parameter Estimation in the Schwartz-Smiths Two-Factor Model," Papers 2108.01881, arXiv.org.
    10. repec:dau:papers:123456789/11429 is not listed on IDEAS
    11. Peilun He & Nino Kordzakhia & Gareth W. Peters & Pavel V. Shevchenko, 2024. "PDSim: A Shiny App for Simulating and Estimating Polynomial Diffusion Models in Commodity Futures," Papers 2409.19385, arXiv.org, revised Oct 2025.

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