Electronic Journal of Statistics

Tail index estimation, concentration and adaptivity

Stéphane Boucheron and Maud Thomas

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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.

Article information

Electron. J. Statist., Volume 9, Number 2 (2015), 2751-2792.

Received: March 2015
First available in Project Euclid: 18 December 2015

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60G70: Extreme value theory; extremal processes 62G30: Order statistics; empirical distribution functions 62G32: Statistics of extreme values; tail inference

Hill estimator adaptivity Lepski’s method concentration inequalities order statistics


Boucheron, Stéphane; Thomas, Maud. Tail index estimation, concentration and adaptivity. Electron. J. Statist. 9 (2015), no. 2, 2751--2792. doi:10.1214/15-EJS1088. https://projecteuclid.org/euclid.ejs/1450456321

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