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 $X^i = (X^i_t)_{0leq tleq T^i}, 1leq ileq N$, 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.
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