Estimation of parameters in stochastic differential equations with two random effects
Keywords: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 (SDEâ€™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.
 Y. AÃ¯tâ€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.
 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.
 S. Beal and L. Shiner, Estimating population kinetics, Critical Reviews in Biomedical Engineering, 8 (1982), 195 - 222.
 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.
 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.
 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.
 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.
 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), 609â€“635.
 R. S. Liptser, and A. N. Shiryaev, Statistics of Random Prcesses I. General Theory, 2nd edition. Springer-Verlag, Berlin, Heidelberg, (2001).
 T. Maitra and S. Bhattacharya, On asymptotic related to clasical inference in stochastic differential equations with random effects, ArXiv: 1407.3968v1, (2014), 1 - 12.
 T. Maitra and S. Bhattacharya. On Bayesian Asymptoics in Stochastic Differential Equations with Random Effects. Statistics and Probability Letters. (2015). to appear. Avaible at â€œhttp://arxiv.org/abs/1407.3971â€.
 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.
 U. Picchini, A. De Gaetano, and S. Ditlevsen, Stochastic differential mixed-effects models, Scand. J. Statist., 37 (2010), 67 â€“ 90. http://dx.doi.org/10.1111/j.1467-9469.2009.00665.x.
 M. J. Schervish, Theory of Statistics, Springer-Verlag, New York. (1995). http://dx.doi.org/10.1007/978-1-4612-4250-5.
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