Tail index estimation, concentration and adaptivity

TL;DR

This paper introduces an adaptive Hill estimator for tail index estimation, leveraging model selection and concentration inequalities to improve accuracy and theoretical guarantees, validated through simulations.

Contribution

It proposes a data-driven, adaptive Hill estimator with oracle inequalities, combining concentration inequalities and Extreme Value Theory tools for improved tail index estimation.

Findings

Estimator satisfies an oracle inequality.

Achieves the lower bound for tail index estimation.

Validated through Monte-Carlo simulations.

Abstract

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 R\'enyi's characterisation of the order statistics of exponential samples. The performance of this computationally and conceptually simple method is illustrated using Monte-Carlo simulations.