Parametric inference for stochastic differential equations with random effects in the drift coefficient

 
 
 
  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract


    In this paper we focus on estimating the parameters in the stochastic differential equations (SDE’s) with drift coefficients depending linearly on a random variables  and  .The distributions of the random effects  and  are depends on unknown parameters from the continuous observations of the independent processes . When  is an unknown parameter or restrict positive constant also studied in this paper. We propose the Gaussian distribution for the random effect  and the exponential distribution for the random effect    , we obtained an explicit formulas for the likelihood functions in each case and find the maximum likelihood estimators of the unknown parameters in the random effects and for the unknown parameter    . Consistency and asymptotic normality are studied just when  is normal random effect and  is constant.


  • Keywords


    Stochastic Differential Equations; Maximum Likelihood Estimator; Linear Random Effects; Fisher Information Matrix; Asymptotic Normality; Consistency.

  • References


      [1] E. Allen, Modeling with Itô stochastic differential equations, Springer Science & Business Media, 2007.

      [2] R. Hindriks, Empirical dynamics of neuronal rhythms, in, Ph. D. Thesis, VU University Amsterdam, 2011.

      [3] M. Musiela, M. Rutkowski, Martingale methods in financial modelling, Springer Science & Business Media, 2006.

      [4] S. Gugushvili, P. Spreij, Parametric inference for stochastic differential equations: a smooth and match approach, arXiv preprint arXiv: 1111. 1120, (2011).

      [5] E. Wong, B. Hajek, Stochastic processes in engineering systems, Springer Science & Business Media, 2012.

      [6] J.N. Nielsen, H. Madsen, P.C. Young, Parameter estimation in stochastic differential equations: an overview, Annual Reviews in Control, 24 (2000) 83-94. http://dx.doi.org/10.1016/S1367-5788(00)90017-8.

      [7] G.B. Durham, A.R. Gallant, Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes, Journal of Business & Economic Statistics, 20 (2002) 297-338. http://dx.doi.org/10.1198/073500102288618397.

      [8] P.D. Feigin, Maximum likelihood estimation for continuous-time stochastic processes, Advances in Applied Probability, (1976) 712-736. http://dx.doi.org/10.1017/S0001867800042890.

      [9] R.V. Overgaard, N. Jonsson, C.W. Tornøe, H. Madsen, Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm, Journal of pharmacokinetics and pharmacodynamics, 32 (2005) 85-107. http://dx.doi.org/10.1007/s10928-005-2104-x.

      [10] C.W. Tornøe, R.V. Overgaard, H. Agersø, H.A. Nielsen, H. Madsen, E.N. Jonsson, Stochastic differential equations in NONMEM®: implementation, application, and comparison with ordinary differential equations, Pharmaceutical research, 22 (2005) 1247-1258. http://dx.doi.org/10.1007/s11095-005-5269-5.

      [11] M. Delattre, 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.

      [12] S. Klim, S.B. Mortensen, N.R. Kristensen, R.V. Overgaard, H. Madsen, Population stochastic modelling (PSM)—an R package for mixed-effects models based on stochastic differential equations, Computer methods and programs in biomedicine, 94 (2009) 279-289. http://dx.doi.org/10.1016/j.cmpb.2009.02.001.0.

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

      [14] U. Picchini, A.D. GAETANO, S. Ditlevsen, Stochastic Differential Mixed‐Effects Models, Scandinavian Journal of statistics, 37 (2010) 67-90. http://dx.doi.org/10.1111/j.1467-9469.2009.00665.x.

      [15] W. Kampowsky, P. Rentrop, W. Schmidt, Classification and numerical simulation of electric circuits, (1992).

      [16] U. Picchini, 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.

      [17] S. Beal, L. Sheiner, Estimating population kinetics, Critical reviews in biomedical engineering, 8 (1981) 195-222.

      [18] A.R. Pedersen, A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations, Scandinavian journal of statistics, (1995) 55-71.

      [19] M.W. Brandt, P. Santa-Clara, Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets, Journal of financial economics, 63 (2002) 161-210. http://dx.doi.org/10.1016/S0304-405X(01)00093-9.

      [20] J. Nicolau, A new technique for simulating the likelihood of stochastic differential equations, The Econometrics Journal, 5 (2002) 91-103. http://dx.doi.org/10.1111/1368-423X.t01-1-00075.

      [21] A. Hurn, K. Lindsay, Estimating the parameters of stochastic differential equations, Mathematics and computers in simulation, 48 (1999) 373-384. http://dx.doi.org/10.1016/S0378-4754(99)00017-8.

      [22] A.W. Lo, Maximum likelihood estimation of generalized Itô processes with discretely sampled data, Econometric Theory, 4 (1988) 231-247. http://dx.doi.org/10.1017/S0266466600012044.

      [23] Y. Aït‐Sahalia, Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed‐form Approximation Approach, Econometrica, 70 (2002) 223-262. http://dx.doi.org/10.1111/1468-0262.00274.

      [24] M. Delattre, V. GENON‐CATALOT, A. Samson, Maximum likelihood estimation for stochastic differential equations with random effects, Scandinavian Journal of Statistics, 40 (2013) 322-343. http://dx.doi.org/10.1111/j.1467-9469.2012.00813.x.

      [25] R. Liptser, A.N. Shiryaev, Statistics of random Processes: I. general Theory, Springer Science & Business Media, 2013.

      [26] A.W. Van der Vaart, Asymptotic statistics, Cambridge university press, 2000.


 

View

Download

Article ID: 6328
 
DOI: 10.14419/ijams.v4i2.6328




Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.