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 DA - 2008/// SP - 196 EP - 218 VL - 12 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007045/ UR - https://www.ams.org/mathscinet-getitem?mr=2374638 UR - https://doi.org/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 https://doi.org/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/

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