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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 1145-1170

\vspace{3pc} \normalsize

\centerline{\large\bf A TWO-STAGE ESTIMATION METHOD FOR RANDOM }
\vspace{2pt} \centerline{\large\bf  COEFFICIENT  DIFFERENTIAL
EQUATION MODELS} \vspace{2pt} \centerline{\large\bf
 WITH APPLICATION TO LONGITUDINAL}
\vspace{2pt} \centerline{\large\bf  HIV DYNAMIC DATA} \vspace{.25cm}
\author{Yun Fang,  Hulin Wu and Li-Xing Zhu}
\vspace{.4cm} \centerline{\it Shanghai Normal University, University
of Rochester } \par
 \centerline{\it and  Hong Kong
Baptist
 University}
\vspace{.55cm} 
\begin{quotation}
\noindent {\it Abstract:} We propose a two-stage estimation method
for random coefficient ordinary differential equation (ODE) models.
A maximum pseudo-likelihood estimator (MPLE) is derived based on a
mixed-effects modeling approach and its asymptotic properties for
population parameters are established. The proposed method does not
require repeatedly solving ODEs, and is computationally efficient
although it does pay a price with the loss of some estimation
efficiency. However, the method does offer an alternative approach
when the exact likelihood approach fails due to model complexity and
high-dimensional parameter space, and it can also serve as a method
to obtain the starting estimates for more accurate estimation
methods. In addition, the proposed method does not need to specify
the initial values of state variables and preserves all the
advantages of the mixed-effects modeling approach. The finite sample
properties of the proposed estimator are studied via Monte Carlo
simulations and the methodology is also illustrated with application
to an AIDS clinical data set.\par \vspace{9pt} \noindent {\it Key
words and phrases:} AIDS/HIV data, local polynomial kernel
smoothing, longitudinal data, mixed-effects models, ordinary
differential equation, pseudo likelihood, viral dynamics.\par
\end{quotation}
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