**Acknowledgements**

is not always possible for a stochastic and continuous process in an SDME model, so the search for an approximation of this density is an important step and requires an expensive calculation. This task is very interesting to give good results with good statistical properties of the estimators obtained by maximizing the likelihood func-

multidimensional and nonlinear SDME models with many random parameters normally distributed that can be extended to random parameters of any distribution.

the Risken approximation for the formal solution of the Fokker-Planck equation proposed by Risken [34], and then the approximated likelihood is obtained using the Laplace approximation method and optimized using the genetic algorithm; these calculation procedures can be obtained using any numerical calculation soft-

The classical inference of SDME models implies the problem of the numerical evaluation of the integral for the given random effects in the likelihood function, which becomes complicated especially when the model contains more than two random parameters. In the literature, several methods have been proposed and tested for the approximation of the integral (see references in the introduction) and the following examples: [8] which proposes the Gaussian quadrature method to solve the integrals for the case of an SDME models with a single random effect and [20] where the study was revised for a general case with several random parameters using the Laplace approximation to compute the integral in Eq. (4) and Eq. (6) numerically. For the mixed effects framework, see [3, 16, 55, 56]. In the case of using the Laplace method, as in this chapter, the calculation of the Hessian matrix can be done analytically when it is possible, as the examples in Section 2, or with the help of a symbolic calculus software or the automatic differentiation (AD)

The results of simulation studies are satisfactory and can be obtained either by using moderate values for the number of experimental units M and of observations n taken for each experimental unit or by using a small sample size but with a number of measurements taken for each subject of 10 at least; this is relevant for applications where large sets of data are not available, such as biomedical applica-

The advantages of this approach, compared to those proposed in the literature for multidimensional SDME models with more than one random parameter [26], are that the computation of the approximate density is very easy and does not require a lot of time to calculate it or to program it in a software; the only task that can be time-consuming is in the optimization step to search for the optimum solution of the likelihood function, and also the proposed method is effective even with large data with a MATLAB program on a common PC (Intel Pentium IV 3.0 GHz with 512 MB of RAM). Nevertheless, the method suffers some limitations, for example, when the conditions to use Eq. (11) are not verified when, e.g., the inverse of the diffusion term does not exist and when, in certain cases, it is not obvious to derive the gradients and Hessians terms. Another limitation is that measurement errors are not considered in this work, and for a good stochastic version, it will be better to include noise on process increments and noise on observations that may be significant compared to system noise. These limitations may provide a perspective towards a more elaborate extension of the statistical

In conclusion, in this work, we proposed a method of parameter estimation for a mixed effects models with SDEs proposing an approximation method for the transition density in the case when it cannot be obtained in a closed form, with an

*<sup>Y</sup>* is obtained in a closed form using

tion. In this work, the proposed estimation method has been applied to

So, an approximation of the transition density *P*ð Þ *<sup>a</sup>*

*Numerical Modeling and Computer Simulation*

ware or with symbolic computing capabilities.

tions where the mixed effect theory is widely applied.

study for SDME models, particularly in the field of epidemiology.

tools [57].

**106**

Z.T., H.E., and F.B conceived the presented idea and contributed to the design and implementation of the research. F.B., H.E., Z.T., and H.A contributed to the analysis of the results. Both F.B. and H.E authors developed the theory and performed the computations and the numerical simulations and verified the analytic methods and calculations.
