Motivated by an analysis of the degree distributions in a large social network, we are concerned with the analysis of heavy-tailed data when a portion of the extreme values are unavailable. We focus on the Hill estimator, which plays a starring role in heavy-tailed modeling. The Hill estimator for this data exhibited a smooth and increasing “sample path” as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation, we introduce a new parameterization into the Hill estimator that corresponds to the proportion of extreme values that are unavailable and the proportion of upper order statistics used in the estimation. We establish functional convergence of the normalized Hill estimator to a Gaussian random field. An estimation procedure is developed based on the limit theory to estimate the number of missing extremes and extreme value parameters including the tail index and the bias of Hill’s estimate. We illustrate how this approach works in both simulations and real data examples.
Gennady Samorodnitsky is a Professor in the School of Operations Research & Industrial Engineering at Cornell University. His research interest lies both in probability theory and in its various applications. He is especially interested in “non-standard” models, in particular those exhibiting heavy tails and/or long-range dependence. His other areas of interest include self-similar (fractal-like) stochastic processes, stable and other infinitely divisible processes, geometry of the excursion sets of random fields, scale free random graphs, and connections between probability and ergodic theory. Professor Samorodnitsky holds an honorary appointment in the Advisory Board of Leibnitz Lab of Insurance and Financial Mathematics in University of Hanover since 2010. His education includes a B.Sc. (in Computer Science) from Moscow Steel and Alloys Institute, a M.Sc. (in Operations Research) and D.Sc. (in Statistics) from Technion – Israel Institute of Technology. He has published over 100 scholarly articles and he is the Associate Editor for Probability and Mathematical Statistics, Stochastic Models, Extremes, Annals of Probability, Applied Probability journals.
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