On Bayesian asymptotics in stochastic differential equations with random effects

Abstract

Delattre et al. (2013) investigated asymptotic properties of the maximum likelihood estimator of the population parameters of the random effects associated with n independent stochastic differential equations (SDE's) assuming that the SDE's are independent and identical (iid).In this article, we consider the Bayesian approach to learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution in the iid set-up as well as when the SDE's are independent but non-identical.

A Short Note on Almost Sure Convergence of Bayes Factors in the General Set-Up

“…Necessary and sufficient conditions for (A3) to hold has more recently been established in (Harrison (2008)). However, in our experience, (A3) is usually easy to verify; see Section S-2 of the supplement; see also Maitra & Bhattacharya (2016b).…”

Inference for Biomedical Data by Using Diffusion Models with Covariates and Mixed Effects

“…The case of continuous time observations of a univariate SDMEM with Gaussian and mixture of Gaussian mixed effects entering the drift linearly was considered in Delattre et al . () and Maitra and Bhattacharya ().…”

On asymptotic inference in stochastic differential equations with time‐varying covariates

“…However, systems of SDE based models consisting of time-varying covariates seem to be rare in the statistical literature, in spite of their importance, and their asymptotic properties are hitherto unexplored. Indeed, although asymptotic inference in single, fixed effects SDE models without covariates has been considered in the literature as time tends to infinity (see, for example, Bishwal (2008)), asymptotic theory in systems of SDE models is rare, and so far only random effects SDE systems without covariates have been considered, as n, the number of subjects (equivalently, the number of SDEs in the system), tends to infinity (Delattre et al (2013), Maitra and Bhattacharya (2016c), Maitra and Bhattacharya (2015)). Such models are of the following form: dX i (t) = b(X i (t), φ i )dt + σ(X i (t))dW i (t), with X i (0) = x i , i = 1, .…”

“…, n}, which are to be interpreted as the random effect parameters associated with the n individuals, which are assumed by Delattre et al (2013) to be independent of the Brownian motions and independently and identically distributed (iid) random variables with some common distribution. For the sake of convenience Delattre et al (2013) (see also Maitra and Bhattacharya (2016c) and Maitra and Bhattacharya (2015)) assume b(x, φ i ) = φ i b(x). Thus, the random effect is a multiplicative factor of the drift function; also, the function b(x) is assumed to be independent of parameters.…”