Improved estimation of the extreme value index using related variables
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Heavy tailed phenomena are naturally analyzed by extreme value statistics. A crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the n observations of interest another n + m observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. We establish the asymptotic normality of this new estimator. It shows greatly improved behavior relative to the Hill estimator, in particular the asymptotic variance is substantially reduced, whereas we can keep the asymptotic bias the same. A simulation study confirms the substantially improved performance of our adapted estimator. We also present an application to the aforementioned earthquake losses.
KeywordsAsymptotic normality Heavy tail Hill estimator Tail dependence Variance reduction
AMS 2000 Subject ClassificationsPrimary—62G32 62G05 62G20 62P05 Secondary—60F05 60G70
We thank an associate editor and three referees for many insightful comments, which helped to improve this paper.
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