A Study Of The Tail Measure And Its Applications In Risk Modeling
This dissertation has 4 chapters, in which we attempt to explore and analyze the structure of extremal data. The first chapter is a review of statistical estimation methods of the tail in the context of extreme value theory as well as their applications in risk managmenet. The quality of estimation of multivariate tails depends significantly on the portion of the sample included in the estimation. Hence, the second chapter describes an approach involving sequential statistical testing in order to select which observations should be used for estimation of the tail. The method is computationally efficient, and can be easily automated. No visual inspection of the data is required. The consistency of the Hill estimator is established when used in conjunction with the proposed method, as well as its asymptotic fluctuations. The third chapter expands the previous method to the multivariate case. The estimator for the tail measure is proven to be consistent using this method of tail selection. We test the proposed method on simulated data, and subsequently apply it to analyze CoVaR for stock and index returns. Finally, we study the structure of spectral measures in financial data. We make observations about certain characteristics of the measures and subsequently propose an approach that can help us study the spectral measure in the face of high dimensional sparsity.
tail index; regular variation; spectral measure
Saloff-Coste, Laurent Pascal; Li, Ping
Ph.D. of Applied Mathematics
Doctor of Philosophy
dissertation or thesis