Multivariate inhomogeneous diffusion models with covariates and mixed effects

TL;DR

This paper introduces a method for approximate maximum-likelihood estimation in multivariate inhomogeneous diffusion models with covariates and mixed effects, enabling flexible subject-specific modeling in longitudinal data analysis.

Contribution

It develops a novel estimation approach for complex multivariate diffusion models with mixed effects, proving consistency and asymptotic normality of the estimators.

Findings

Estimator is consistent and asymptotically normal as the number of subjects increases.

Bias from time-discretization of sufficient statistics is analyzed.

Simulation studies demonstrate the method's effectiveness.

Abstract

Modeling of longitudinal data often requires diffusion models that incorporate overall time-dependent, nonlinear dynamics of multiple components and provide sufficient flexibility for subject-specific modeling. This complexity challenges parameter inference and approximations are inevitable. We propose a method for approximate maximum-likelihood parameter estimation in multivariate time-inhomogeneous diffusions, where subject-specific flexibility is accounted for by incorporation of multidimensional mixed effects and covariates. We consider $N$ multidimensional independent diffusions $X^i = (X^i_t)_{0\leq t\leq T^i}, 1\leq i\leq N$, with common overall model structure and unknown fixed-effects parameter $\mu$. Their dynamics differ by the subject-specific random effect $\phi^i$ in the drift and possibly by (known) covariate information, different initial conditions and observation times and duration. The distribution of $\phi^i$ is parametrized by an unknown $\vartheta$ and $\theta = (\mu, \vartheta)$ is the target of statistical inference. Its maximum likelihood estimator is derived from the continuous-time likelihood. We prove consistency and asymptotic normality of $\hat{\theta}_N$ when the number $N$ of subjects goes to infinity using standard techniques and consider the more general concept of local asymptotic normality for less regular models. The bias induced by time-discretization of sufficient statistics is investigated. We discuss verification of conditions and investigate parameter estimation and hypothesis testing in simulations.