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*Duke Economics Working Paper #96-03*

# The Nonlinear Mixed Effects Model with a Smooth Random Effects
Density

##
Marie Davidian

and

A. Ronald Gallant

### Abstract

The fixed parameters of the nonlinear mixed effects model and the
density of the random effects are estimated jointly by maximum
likelihood. The density of the random effects is assumed to be
smooth but is otherwise unrestricted. The method uses a series
expansion that follows from the smoothness assumption to
represent the density and quadrature to compute the likelihood.
Standard algorithms are used for optimization. Empirical Bayes
estimates of random coefficients are obtained by computing
posterior modes. The method is applied to data from
pharmacokinetics, and properties of the method are investigated
by application to simulated data.
Key Words: Maximum likelihood; nonlinear mixed effects models;
nonparametric; pharmacokinetics.

Published in *Biometrika*, Vol. 80, 1993, pp. 474-488.