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Sparse functional/longitudinal data have attracted widespread interest due tothe prevalence of such data in social and life sciences. A prominent scenariowhere such data are routinely encountered are accelerated longitudinal studies,where subjects are enrolled in the study at a random time and are only trackedfor a short amount of time relative to the domain of interest. The statisticalanalysis of such functional snippets is challenging since information for thefar-off-diagonal regions of the covariance structure is missing. Our mainmethodological contribution is to address this challenge by bypassingcovariance estimation and instead modeling the underlying process as thesolution of a data-adaptive stochastic differential equation. Taking advantageof the interface between Gaussian functional data and stochastic differentialequations makes it possible to efficiently reconstruct the target process byestimating its dynamic distribution. The proposed approach allows one toconsistently recover forward sample paths from functional snippets at thesubject level. We establish the existence and uniqueness of the solution to theproposed data-driven stochastic differential equation and derive rates ofconvergence for the corresponding estimators. The finite-sample performance isdemonstrated with simulation studies and functional snippets arising from agrowth study and spinal bone mineral density data.

Dynamic Modeling of Sparse Longitudinal Data and Functional Snippets With Stochastic Differential Equations