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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 1831-1856

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2009.328

\vspace{3pc} \normalsize \centerline{\large\bf VARYING COEFFICIENT
MODELS FOR SPARSE } \vspace{2pt} \centerline{\large\bf
NOISE-CONTAMINATED LONGITUDINAL DATA} \vspace{.25cm}
\author{Damla \c{S}ent\"{u}rk and Danh V. Nguyen}
\vspace{.4cm} \centerline{\it University of California, Los Angeles
and University of California, Davis} \vspace{.55cm} 
\begin{quotation}
\noindent {\it Abstract:} In this paper we propose a varying
coefficient model for sparse longitudinal data that allows for
error-prone time-dependent variables and time-invariant covariates.
We develop a new estimation procedure, based on covariance
representation techniques, that enables effective borrowing of
information across all subjects in sparse and irregular longitudinal
data observed with measurement error, a challenge for which there is
no current adequate solution. Sparsity is addressed via a functional
analysis approach that considers the observed longitudinal data as
noise contaminated realizations of a random process that produces
smooth trajectories. This approach allows for estimation based on
pooled data, borrowing strength from all subjects, in targeting the
mean functions and auto- and cross-covariances to overcome sparse
noisy designs. The resulting estimators are shown to be uniformly
consistent. Consistent prediction for the response trajectories  are
also obtained via conditional expectation under Gaussian
assumptions. Asymptotic distributions of the predicted response
trajectories are derived, allowing for construction of asymptotic
pointwise confidence bands. Efficacy of the proposed method is
investigated
 in simulation studies and compared to the commonly
used local polynomial smoothing method. The proposed method is
illustrated with a sparse longitudinal data set, examining the
age-varying relationship between calcium absorption and dietary
calcium. Prediction of individual calcium absorption curves as a
function of age are also examined. \par \vspace{9pt} \noindent {\it
Key words and phrases:} Functional data analysis, local least
squares, measurement error, repeated measurements, smoothing, sparse
design. \par
\end{quotation}

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