Non-linear mixed models defined by stochastic differential equations (SDEs) are considered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measurement instants. A tuned hybrid Gibbs algorithm based on conditional brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach.

Keywords: brownian bridge, diffusion process, Euler-Maruyama approximation, Gibbs algorithm, incomplete data model, maximum likelihood estimation, non-linear mixed effects model, SAEM algorithm

@article{PS_2008__12__196_0, author = {Donnet, Sophie and Samson, Adeline}, title = {Parametric inference for mixed models defined by stochastic differential equations}, journal = {ESAIM: Probability and Statistics}, pages = {196--218}, publisher = {EDP-Sciences}, volume = {12}, year = {2008}, doi = {10.1051/ps:2007045}, mrnumber = {2374638}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2007045/} }

TY - JOUR AU - Donnet, Sophie AU - Samson, Adeline TI - Parametric inference for mixed models defined by stochastic differential equations JO - ESAIM: Probability and Statistics PY - 2008 SP - 196 EP - 218 VL - 12 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007045/ DO - 10.1051/ps:2007045 LA - en ID - PS_2008__12__196_0 ER -

%0 Journal Article %A Donnet, Sophie %A Samson, Adeline %T Parametric inference for mixed models defined by stochastic differential equations %J ESAIM: Probability and Statistics %D 2008 %P 196-218 %V 12 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2007045/ %R 10.1051/ps:2007045 %G en %F PS_2008__12__196_0

Donnet, Sophie; Samson, Adeline. Parametric inference for mixed models defined by stochastic differential equations. ESAIM: Probability and Statistics, Volume 12 (2008), pp. 196-218. doi : 10.1051/ps:2007045. http://www.numdam.org/articles/10.1051/ps:2007045/

[1] Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70 (2002) 223-262. | MR | Zbl

,[2] On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Prob. 16 (2006) 1462-1505. | MR | Zbl

and ,[3] The law of the Euler Scheme for Stochastic Differential Equations: I. Convergence Rate of the Density. Technical Report 2675, INRIA (1995). | MR

and ,[4] The law of the Euler scheme for stochastic differential equations (II): convergence rate of the density. Monte Carlo Methods Appl. 2 (1996) 93-128. | MR | Zbl

and ,[5] Estimating population kinetics. Crit. Rev. Biomed. Eng. 8 (1982) 195-222.

and ,[6] MCMC for nonlinear hierarchical models. Chapman & Hall, London (1996) 339-358.

, and ,[7] Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. B 68 (2006) 333-382. | MR | Zbl

, , and ,[8] Exact simulation of diffusions. Ann. Appl. Prob. 15 (2005) 2422-2444. | MR | Zbl

and ,[9] Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1 (1995) 17-39. | MR | Zbl

and ,[10] The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Computational. Statistics Quaterly 2 (1985) 73-82.

and ,[11] Probabilités et statistiques. Tome 2. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree]. Masson, Paris, 1983. Problèmes à temps mobile. [Movable-time problems]. | MR | Zbl

and ,[12] Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19 (1986) 263-284. | MR | Zbl

and ,[13] Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 (1999) 94-128. | MR | Zbl

, and ,[14] Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm. Stoch. Process. Appl. 23 (1986) 91-113. | MR | Zbl

and ,[15] A P. Dempster, N.M. Laird and D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 (1977) 1-38. With discussion. | MR | Zbl

[16] Mixed effects in stochastic differential equation models. REVSTAT- Statistical Journal 3 (2005) 137-153. | MR | Zbl

and ,[17] Estimation of parameters in incomplete data models defined by dynamical systems. J. Stat. Plan. Inf. (2007). | MR

and ,[18] Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 (2001) 381-420. | MR | Zbl

and ,[19] Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 (2001) 959-993. | MR | Zbl

, and ,[20] MCMC analysis of diffusion models with application to finance. J. Bus. Econ. Statist. 19 (2001) 177-191. | MR

,[21] On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 119-151. | Numdam | MR | Zbl

and ,[22] Diffusions with measurement errors. I. Local asymptotic normality. ESAIM: PS 5 (2001) 225-242. | Numdam | MR | Zbl

and ,[23] Diffusions with measurement errors. II. Optimal estimators. ESAIM: PS 5 (2001) 243-260 (electronic). | Numdam | MR | Zbl

and ,[24] Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist. 24 (1997) 211-229. | MR | Zbl

,[25] Applications of Pharmacokinetic principles in drug development. Kluwer Academic/Plenum Publishers, New York (2004).

,[26] Coupling a stochastic approximation version of EM with a MCMC procedure. ESAIM: PS 8 (2004) 115-131. | Numdam | MR | Zbl

and ,[27] Maximum likelihood estimation in nonlinear mixed effects models. Comput. Statist. Data Anal. 49 (2005) 1020-1038. | MR

and ,[28] Applications of the malliavin calculus, part II. J. Fac. Sci. Univ. Tokyo. Sect. IA, Math. 32 (1985) 1-76. | MR | Zbl

and ,[29] Parameter estimation for stochastic processes. Helderman Verlag Berlin (1984). | MR | Zbl

,[30] Nonlinear mixed effects models for repeated measures data. Biometrics 46 (1990) 673-687. | MR

and ,[31] Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44 (1982) 226-233. | MR | Zbl

,[32] Non-linear mixed-effects models with stochastic differential equations: Implementation of an estimation algorithm. J Pharmacokinet. Pharmacodyn. 32 (2005) 85-107.

, , and ,[33] A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist. 22 (1995) 55-71. | MR | Zbl

,[34] Approximations to the log-likelihood function in the non-linear mixed-effect models. J. Comput. Graph. Statist. 4 (1995) 12-35.

and ,[35] Approximate maximum likelihood estimation of discretely observed diffusion process. Center for Analytical Finance, Working paper 29 (1999).

,[36] Statistical Inference for Diffusion Type Processes. Arnold Publisher (1999). | MR | Zbl

,[37] On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika 88 (2001) 603-621. | MR | Zbl

and ,[38] Evaluation of likelihood function for gaussian signals. IEEE Trans. Inf. Theory 11 (1965) 61-70. | MR | Zbl

,[39] Continuous-time dynamical systems with sampled data, error of measurement and unobserved components. J. Time Series Anal. 14 (1993) 527-545. | MR | Zbl

,[40] Parametric inference for diffusion processes observed at discrete points in time: a survey. Int. Stat. Rev 72 (2004) 337-354.

,[41] Prediction-based estimating functions. Econom. J. 3 (2000) 123-147. | MR | Zbl

,[42] Markov chains for exploring posterior distributions. Ann. Statist. 22 (1994) 1701-1762. | MR | Zbl

,[43] Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations. Pharm. Res. 22 (2005) 1247-1258.

, , , , and ,[44] Calculating the content and boundary of the highest posterior density region via data augmentation. Biometrika 77 (1990) 649-652. | MR

and ,[45] Laplace's approximation for nonlinear mixed models. Biometrika 80 (1993) 791-795. | MR | Zbl

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