Estimation for stochastic differential equation mixed models using approximation methods

We used a class of stochastic differential equations (SDE) to model the evolution of cattle weight that, by an appropriate transformation of the weight, resulted in a variant of the Ornstein-Uhlenbeck model. In previous works, we have dealt with estimation, prediction, and optimization issues for this class of models. However, to incorporate individual characteristics of the animals, the average transformed size at maturity parameter $\alpha$ and/or the growth parameter $\beta$ may vary randomly from animal to animal, which results in SDE mixed models. Obtaining a closed-form expression for the likelihood function to apply the maximum likelihood estimation method is a difficult, sometimes impossible, task. We compared the known Laplace approximation method with the delta method to approximate the integrals involved in the likelihood function. These approaches were adapted to allow the estimation of the parameters even when the requirement of most existing methods, namely having the same age vector of observations for all trajectories, fails, as it did in our real data example. Simulation studies were also performed to assess the performance of these approximation methods. The results show that the approximation methods under study are a very good alternative for the estimation of SDE mixed models.