Open Access
2015 Tail index estimation, concentration and adaptivity
Stéphane Boucheron, Maud Thomas
Electron. J. Statist. 9(2): 2751-2792 (2015). DOI: 10.1214/15-EJS1088


This paper presents an adaptive version of the Hill estimator based on Lespki’s model selection method. This simple data-driven index selection method is shown to satisfy an oracle inequality and is checked to achieve the lower bound recently derived by Carpentier and Kim. In order to establish the oracle inequality, we derive non-asymptotic variance bounds and concentration inequalities for Hill estimators. These concentration inequalities are derived from Talagrand’s concentration inequality for smooth functions of independent exponentially distributed random variables combined with three tools of Extreme Value Theory: the quantile transform, Karamata’s representation of slowly varying functions, and Rényi’s characterisation for the order statistics of exponential samples. The performance of this computationally and conceptually simple method is illustrated using Monte-Carlo simulations.


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Stéphane Boucheron. Maud Thomas. "Tail index estimation, concentration and adaptivity." Electron. J. Statist. 9 (2) 2751 - 2792, 2015.


Received: 1 March 2015; Published: 2015
First available in Project Euclid: 18 December 2015

zbMATH: 1352.60025
MathSciNet: MR3435810
Digital Object Identifier: 10.1214/15-EJS1088

Primary: 60E15 , 60G70 , 62G30 , 62G32

Keywords: Adaptivity , Concentration inequalities , Hill estimator , Lepski’s method , order statistics

Rights: Copyright © 2015 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.9 • No. 2 • 2015
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