Methods for extremes of functional data - Xiong, Qian, author; Kokoszka, Piotr S., advisor; Cooley, Daniel, committee member; Pinaud, Olivier, committee member; Wang, Haonan, committee member (Colorado State University. Libraries, 2018)
Motivated by the problem of extreme behavior of functional data, we develop statistical theory at the nexus of functional data analysis (FDA) and extreme value theory (EVT). A fundamental technique of functional data analysis is to replace infinite dimensional curves with finite dimensional representations in terms of functional principal components (FPCs). The coefficients of these projections, called the scores, encode the shapes of the curves. Therefore, the study of the extreme behavior of functional time series can be transformed to the study on functional principal component scores. We first derive two tests of significance of the slope function using functional principal components and their empirical counterparts (EFPC's). Applied to tropical storm data, these tests show a significant trend in the annual pattern of upper wind speed levels of hurricanes. Then we establish sufficient conditions under which the asymptotic extreme behavior of the multivariate estimated scores is the same as that of the population scores. We clarify these issues, including the rate of convergence, for Gaussian functions and for more general functional time series whose projections are in the Gumbel domain of attraction. Finally, we derive the asymptotic distribution of the sample covariance operator and of the sample functional principal components for functions which are regularly varying and whose fourth moment does not exist. The new theory is applied to establish the consistency of the regression operator in a functional linear model, with such errors.