On Classical and Bayesian Asymptotics in Stochastic Differential Equations with Random Effects having Mixture Normal Distributions

Delattre et al. (2013) considered a system of stochastic differential equations (SDE's) in a random effects set-up. Under the independent and identical (iid) situation, and assuming normal distribution of the random effects, they established weak consistency of the maximum likelihood estimators (MLE's) of the population parameters of the random effects. In this article, respecting the increasing importance and versatility of normal mixtures and their ability to approximate any standard distribution, we consider the random effects having finite mixture of normal distributions and prove asymptotic results associated with the MLE's in both independent and identical (iid) and independent but not identical (non-iid) situations. Besides, we consider iid and non-iid set-ups under the Bayesian paradigm and establish posterior consistency and asymptotic normality of the posterior distribution of the population parameters. It is important to note that Delattre et al. (2016) also assumed the SDE set-up with normal mixture distribution of the random effect parameters but considered only the iid case and proved only weak consistency of the MLE under an extra, strong assumption as opposed to strong consistency that we are able to prove without the extra assumption. Furthermore, they did not deal with asymptotic normality of MLE or the Bayesian asymptotics counterpart which we investigate in details.