The paper is concerned with the extreme behavior of projections of time series of functions onto data-driven basis systems, for example, on the estimated functional principal components. The coefficients of these projections, called the scores, encode the shapes of the curves. Within the framework of functional data analysis, the extreme shapes are those corresponding to multivariate extremes of the scores. The scores are not directly observable, and must be computed from the data. Even for iid Gaussian functions, they form a triangular array of dependent non–Gaussian vectors. Thus, even though the extreme behavior of the population scores of Gaussian functions follows from well–known results, it is not clear what the extreme behavior of their approximations computed from the data is. We clarify these issues for Gaussian functions and for more general functional time series whose projections are in the Gumbel domain of attraction.
Extreme values Functional data Gumbel distribution Time series
AMS 2000 Subject Classification
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This research was partially supported by NSF grant DMS 1462067 “FRG: Collaborative Research: Extreme Value Theory for Spatially Indexed Functional Data”. We thank Professor Haonan Wang for useful advice on elements of the proof in the Gaussian case, and two referees for valuable advice on both presentation and substance.
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