Kim, Mihyun, authorKokoszka, Piotr, advisorCooley, Daniel, committee memberMeyer, Mary, committee memberPinaud, Olivier, committee member2021-09-062021-09-062021https://hdl.handle.net/10217/233780This dissertation is concerned with the asymptotic theory of statistical tools used in extreme value analysis of functional data and internet anomaly data. More specifically, we study four problems associated with analyzing the tail behavior of functional principal component scores in functional data and interarrival times of internet traffic anomalies, which are available only with a round-off error. The first problem we consider is the estimation of the tail index of scores in functional data. We employ the Hill estimator for the tail index estimation and derive conditions under which the Hill estimator computed from the sample scores is consistent for the tail index of the unobservable population scores. The second problem studies the dependence between extremal values of functional scores using the extremal dependence measure (EDM). After extending the EDM defined for positive bivariate observations to multivariate observations, we study conditions guaranteeing that a suitable estimator of the EDM based on these scores converges to the population EDM and is asymptotically normal. The third and last problems investigate the asymptotic and finite sample behavior of the Hill estimator applied to heavy-tailed data contaminated by errors. For the third one, we show that for time series models often used in practice, whose non–contaminated marginal distributions are regularly varying, the Hill estimator is consistent. For the last one, we formulate conditions on the errors under which the Hill and Harmonic Moment estimators applied to i.i.d. data continue to be asymptotically normal. The results of large and finite sample investigations are applied to internet anomaly data.born digitaldoctoral dissertationsengCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.functional data analysisheavy tail analysisextremal dependence measureHill estimatorfunctional principal component scoresText