ABSTRACT

This research project provides simultaneous confidence regions for various functional features in functional data analysis (FDA), with asymptotic theory and guide to practical implementation. Specifically, asymptotically correct confidence regions will be constructed for (1) the mean function of functional data and the coefficient function in varying coefficient longitudinal regression model; and (2) the covariance function of functional data and the regression function in functional linear model. For the simpler functions in (1), the investigator will employ both regression spline and local polynomial methods in order to establish rigorous asymptotic theory for both sparse and dense function data. Results on partial sum strong approximation by Brownian motions and advanced extreme value theory for sequences of non-stationary Gaussian processes will be applied to obtain distributional properties of the maximal deviation processes. For the more complicated functions in (2), the investigator will propose two-step estimators and show that it is asymptotically as efficient as some ?infeasible? analogs. Asymptotic distributions for maximal deviations are established for the ?infeasible estimators? which are then inherited by the two-step estimators.

Functional data, also known as curve data, consist of collections of digitally recorded curves or surfaces, often with random errors. Such data abound in virtually all scientific disciplines, including but not limited to, climatology, clinical studies, epidemiology, evolutionary biology and food engineering/science. The need to draw information out of a sample of curves, coupled with the unleashing of modern computing power, has made functional data analysis (FDA) one of the most active areas of contemporary statistics research. While multivariate statistics is about unknown vectors and matrices, FDA concerns unknown curves and surfaces, which is most naturally done with confidence regions. The methods developed by the investigator fill a major gap in the current FDA methodology, which lacks procedures to make conclusions on an entire curve with quantifiable uncertainty. Codes written in common software packages such as Matlab or R will be freely distributed so practitioners from academia and industry for analyzing functional data in real time, with own chosen significance levels. Completing this project depends crucially on several capable Ph. D. students working under the investigator?s supervision, so state-of-the-art research is integrated with the training of graduate students as future researchers, consistent with NSF's education goal.

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