Estimation of parameters in stochastic differential equations with two random effects

  • Authors

    • Mohammed Alsukaini huazhong university
    • Walaa Alkreemawi huazhong university
    • Xiang-Jun Wang huazhong university
    2016-04-18
    https://doi.org/10.14419/ijamr.v5i2.5996
  • Stochastic Differential Equations, Maximum Likelihood Estimator, Nonlinear Random Effects, Posterior Consistency, Posterior Normality.
  • In this paper we investigate consistency and asymptotic normality of the posterior distribution of the parameters in the stochastic differential equations (SDE’s) with diffusion coefficients depending nonlinearly on a random variables  and  (the random effects).The distributions of the random effects  and  depends on unknown parameters which are to be estimated from the continuous observations of the independent processes . We propose the Gaussian distribution for the random effect  and the exponential distribution for the random effect    , we obtained an explicit formula for the likelihood function and find the estimators of the unknown parameters in the random effects.

  • References

    1. [1] Y. Aïtâ€Sahalia, Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closedâ€form Approximation Approach, Econometrica 70, no. 1 (2002), 223-262. http://dx.doi.org/10.1111/1468-0262.00274.

      [2] W.K, Alkreemawi, M. S. Alsukaini, and X.J. Wang, On Parameters Estimation in Stochastic Differential Equations with Additive Random Effects, journal of advances in mathematics, 11, no.3 (2015), 5018-5028.

      [3] S. Beal and L. Shiner, Estimating population kinetics, Critical Reviews in Biomedical Engineering, 8 (1982), 195 - 222.

      [4] M. Delattre and M. Lavielle, Coupling the SAEM algorithm and the extended Kalman filter for maximum likelihood estimation in mixed-effects diffusion models, Statistics and Its Interface, 6 (2013), 519 - 532. http://dx.doi.org/10.4310/SII.2013.v6.n4.a10.

      [5] M. Delattre, V. Genon-Catalot, and A. Samson, Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient, Preprint MAP, 5 (2014), 2014 - 07.

      [6] M. Delattre, V. Genon-Catalot, and A. Samson, Maximum likelihood estimation for stochastic differential equations with random effects, Scandinavian Journal of Statistics, 40 (2012), 322 - 343. http://dx.doi.org/10.1111/j.1467-9469.2012.00813.x.

      [7] S. Donnet and A. Samson, A review on estimation of stochastic differential equations for pharmacokinetic-pharmacodynamics models, Advanced Drug Delivery Reviews, 65 (2013), 929 - 939. http://dx.doi.org/10.1016/j.addr.2013.03.005.

      [8] S. Gugushvili and P. Spreij, Parametric inference for stochastic differential equations: a smooth and match approach, ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2), (2012), 609–635.

      [9] R. S. Liptser, and A. N. Shiryaev, Statistics of Random Prcesses I. General Theory, 2nd edition. Springer-Verlag, Berlin, Heidelberg, (2001).

      [10] T. Maitra and S. Bhattacharya, On asymptotic related to clasical inference in stochastic differential equations with random effects, ArXiv: 1407.3968v1, (2014), 1 - 12.

      [11] T. Maitra and S. Bhattacharya. On Bayesian Asymptoics in Stochastic Differential Equations with Random Effects. Statistics and Probability Letters. (2015). to appear. Avaible at “http://arxiv.org/abs/1407.3971â€.

      [12] U. Picchini and S. Ditlevsen, Practical estimation of high dimensional stochastic differential mixed-effects models, Computational Statistics & Data Analysis, 55 (2011), 1426 - 1444. http://dx.doi.org/10.1016/j.csda.2010.10.003.

      [13] U. Picchini, A. De Gaetano, and S. Ditlevsen, Stochastic differential mixed-effects models, Scand. J. Statist., 37 (2010), 67 – 90. http://dx.doi.org/10.1111/j.1467-9469.2009.00665.x.

      [14] M. J. Schervish, Theory of Statistics, Springer-Verlag, New York. (1995). http://dx.doi.org/10.1007/978-1-4612-4250-5.

  • Downloads

  • How to Cite

    Alsukaini, M., Alkreemawi, W., & Wang, X.-J. (2016). Estimation of parameters in stochastic differential equations with two random effects. International Journal of Applied Mathematical Research, 5(2), 97-102. https://doi.org/10.14419/ijamr.v5i2.5996