A review on estimation methods of NonLinear Mixed Effects model with Stochastic Differential Equations, application to threedimensional Ornstein-Uhlenbeck process.

Abstract

In this paper, we focus on estimation methods for non-linear mixed effects (NLME) models with stochastic differential equations (SDEs). This type of model is very useful in multidisciplinary research since it allows to adequately model different phenomena by taking into account their dynamic sides based the two main ingredients: stochastic differential equations and the random effects. As a result, it is possible to take into account different factors that would otherwise be difficult to include in the model. The resulting modelling is further enhanced by the population approach where we consider a whole population of subjects simultaneously rather than a single individual as in the standard approaches. With these advantages in mind, it becomes obvious that that there is a need for both theoretical and practical tools for manipulate such models. Unfortunately, statistical inference in this area remains a complicated task, whereas it seems useful to make an overview of the current state of knowledge on various existing methods which have proven to be most fruitful and effective. Therefore, we introduce these different methods according to the different issues related to this type of modelling, depending on the availability of the density of transitions in explicit form and the existence or not of measurement noise when the process is observed directly or indirectly. Finally, we illustrate this review of estimation methods for non-linear SDME models by the implementation of a simulation study and estimation of the three-dimensional Ornstein-Uhlenbeck process (OU), in its multidimensional and stochastic version.

Keywords

• NLME
• SDE
• Brownian motion
• OU-process
• Maximum likelihood estimation
• Population approach.