Parametric inference for mixed models defined by stochastic differential equations

Sophie Donnet; Adeline Samson

ESAIM: Probability and Statistics (2008)

  • Volume: 12, page 196-218
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Donnet, Sophie, and Samson, Adeline. "Parametric inference for mixed models defined by stochastic differential equations." ESAIM: Probability and Statistics 12 (2008): 196-218. <http://eudml.org/doc/250428>.

@article{Donnet2008,
abstract = { 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. },
author = {Donnet, Sophie, Samson, Adeline},
journal = {ESAIM: Probability and Statistics},
keywords = {Brownian bridge; diffusion process; Euler-Maruyama approximation; Gibbs algorithm; incomplete data model; maximum likelihood estimation; non-linear mixed effects model; SAEM algorithm},
language = {eng},
month = {1},
pages = {196-218},
publisher = {EDP Sciences},
title = {Parametric inference for mixed models defined by stochastic differential equations},
url = {http://eudml.org/doc/250428},
volume = {12},
year = {2008},
}

TY - JOUR
AU - Donnet, Sophie
AU - Samson, Adeline
TI - Parametric inference for mixed models defined by stochastic differential equations
JO - ESAIM: Probability and Statistics
DA - 2008/1//
PB - EDP Sciences
VL - 12
SP - 196
EP - 218
AB - 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.
LA - eng
KW - Brownian bridge; diffusion process; Euler-Maruyama approximation; Gibbs algorithm; incomplete data model; maximum likelihood estimation; non-linear mixed effects model; SAEM algorithm
UR - http://eudml.org/doc/250428
ER -

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