PhD Candidate, Statistics
Extreme value theory is a branch of statistics that is devoted to studying the phenomena governed by extremely rare events. The modeling and statistics of such phenomena are tail dependent and so we consider a class of heavy-tail distributions, which are characterized by regular variation in the tails. While many articles have considered regular variation at one endpoint (particularly the left endpoint), the idea of regular variation at both endpoints has not be addressed. In this dissertation, we propose extreme value estimators for various non-negative time series, where the second or higher moments does not exist and the innovations are positive random variables with regular variation at both the right endpoint infinity and the positive left endpoint θ. This contrasts with traditional estimators whose asymptotic behavior depends on the central part of the innovation distribution. For certain estimation problems like this one, the presence of heavy tails can provide the setting for exceedingly accurate estimates. Within each model, we provide estimates for the model parameters with respect to an extreme value criteria. Through the use of regular variation and point processes the limit distribution for the proposed estimators are obtained while weak convergence results for asymptotically independent joint distribution is derived. A simulation study was performed to first assess the small sample size behavior and reliability of our proposed estimates and secondly to compare the performance of our extreme value estimation procedure and that of traditional and alternative estimation procedures. The main goal of all proposed methods, is to capitalize on the behavior of extreme value estimators over traditional estimators when the regular varying exponent β is 0 < β < 2. In this heavy-tail regime, extreme value estimators converge at a rate faster than square root n. If a practitioner can entertain an infinite variance time series model, then methods such as the one proposed in this dissertation should receive consideration and even more so if an infinite mean time series model were deemed to be acceptable.