  Longitudinal data are often plagued with sparsity of time points where
measurements are available. The functional data analysis perspective has been
shown to provide an effective and flexible approach to address this problem for
the case where measurements are sparse but their times are randomly distributed
over an interval. Here we focus on a different scenario where available data
can be characterized as snippets, which are very short stretches of
longitudinal measurements. For each subject the stretch of available data is
much shorter than the time frame of interest, a common occurrence in
accelerated longitudinal studies. An added challenge is introduced if a time
proxy that is basic for usual longitudinal modeling is not available. This
situation arises in the case of Alzheimer's disease and comparable scenarios,
where one is interested in time dynamics of declining performance, but the time
of disease onset is unknown and the chronological age does not provide a
meaningful time reference for longitudinal modeling. Our main methodological
contribution is to address this problem with a novel approach. Key quantities
for our approach are conditional quantile trajectories for monotonic processes
that emerge as solutions of a dynamic system, and for which we obtain uniformly
consistent estimates. These trajectories are shown to be useful to describe
processes that quantify deterioration over time, such as hippocampal volumes in
Alzheimer's patients.
