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EXTREME VALUE ANALYSIS WITHOUT THE LARGEST VALUES: WHAT CAN BE DONE?

Published online by Cambridge University Press:  30 January 2019

Jingjing Zou Open the ORCID record for Jingjing Zou [Opens in a new window] ,
Richard A. Davis  and
Gennady Samorodnitsky
Jingjing Zou
Affiliation:
Department of Statistics Columbia University, New York, NY, USA E-mail: jz2335@columbia.edu; rdavis@stat.columbia.edu
Richard A. Davis
Affiliation:
Department of Statistics Columbia University, New York, NY, USA E-mail: jz2335@columbia.edu; rdavis@stat.columbia.edu
Gennady Samorodnitsky
Affiliation:
School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA E-mail: gs18@cornell.edu
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Abstract

In this paper, we are concerned with the analysis of heavy-tailed data when a portion of the extreme values is unavailable. This research was motivated by an analysis of the degree distributions in a large social network. The degree distributions of such networks tend to have power law behavior in the tails. We focus on the Hill estimator, which plays a starring role in heavy-tailed modeling. The Hill estimator for these 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 version of the Hill estimator. It is a function of the number of the upper order statistics used in the estimation, but also depends on the number of unavailable extreme values. We establish functional convergence of the normalized Hill estimator to a Gaussian process. 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 estimator. We illustrate how this approach works in both simulations and real data examples.

Keywords

functional convergenceheavy-tailed distributionshill estimatormissing extremes
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 2 , April 2020 , pp. 200 - 220
Copyright
Copyright © Cambridge University Press 2019

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