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

TL;DR

This paper extends asymptotic analysis of maximum likelihood and Bayesian estimators for stochastic differential equations with random effects, allowing for mixture normal distributions and both iid and non-iid data, with proofs of consistency and normality.

Contribution

It introduces asymptotic results for MLEs and Bayesian posteriors in SDE models with mixture normal random effects, including non-iid cases and unknown mixture components.

Findings

Proves strong consistency of MLEs without extra assumptions

Establishes asymptotic normality of MLEs and Bayesian posteriors

Demonstrates effectiveness through simulations and real data

Abstract

Delattre et al. (2013) considered a system of stochastic differential equations (SDEs) in a random effects setup. Under the independent and identical (iid) situation, and assuming normal distribution of the random effects, they established weak consistency of the maximum likelihood estimators (M LEs) 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 mixture of normal distributions and prove asymptotic results associated with the MLEs in both independent and identical (iid) and independent but not identical (non-iid) situations. Besides, we consider iid and non-iid setups under the Bayesian paradigm and establish posterior consistency and asymptotic normality of the posterior distribution of the population parameters, even when the number of mixture components is unknown and treated as a random variable. Although ours is an independent work, we later noted that Delattre et al. (2016) also assumed the SDE setup with normal mixture distribution of the random effect parameters but considered only the iid case and proved only weak consistency of the M LE 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 M LE or the Bayesian asymptotics counterpart which we investigate in details. Ample simulation experiments and application to a real, stock market data set reveal the importance and usefulness of our methods even for small samples.