Parameter Estimation in Stochastic Differential Mixed-Effects
Models
U. Picchini, A. De Gaetano and S. Ditlevsen
Published as
Research Report 06/12, Department of Biostatistics, University of Copenhagen, 2006.
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research report (824 Kb, pdf). Notice that you might be interested in a newer and improved version of the methods considered in the research report, available here.
Stochastic differential equation (SDE) models have shown useful to describe continuous
time processes, e.g. a physiological process evolving in an individual. Biomedical
experiments often imply repeated measurements on a series of individuals or experimental
units and individual differences can be represented by incorporating random effects into
the model. When both system noise and individual differences are considered, stochastic
differential mixed effects models ensue. In most cases the likelihood function is not
available, and thus maximum likelihood estimation is not possible. Here we propose to
approximate the unknown likelihood function by first approximating the conditional transition
density of the diffusion process given the random effects by a Hermite expansion,
as suggested by Ait-Sahalia (2001, 2002), and then numerically integrate the obtained
conditional likelihood with respect to the random effects. The approximated maximum
likelihood estimators are evaluated on simulations from the Ornstein-Uhlenbeck process
and Geometric Brownian motion.
Keywords: Approximate maximum likelihood; closed-form transition density expansion;
Hermite expansion; random effects; Ornstein-Uhlenbeck process; Geometric Brownian motion;
diffusion processes; stochastic differential equations; biomedical applications.
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