Abstract
Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov’s central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill estimators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly right-truncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained.
Highlights
Introduction and motivationHeavy-tailed random variables appear in numerous fields of statistical applications of extreme value analysis, such as insurance and finance (see e.g. p.9 of Embrechts et al (1997)), geoscience (see Section 1.3.5 of Beirlant et al (2004)) and analysis of teletraffic data (see Section 8 of Resnick (2007))
A well-established procedure for extreme quantile estimation is Weissman’s extrapolation method (Weissman 1978), whose essential requirement is a consistent estimator of the tail index of the underlying heavy-tailed distribution
Recent studies geared towards insurance and finance have been advocating for the use and estimation of extreme versions of alternatives to quantiles, such as Conditional Tail Moments (El Methni et al 2014), Wang distortion risk measures (introduced in Wang (1996), and studied recently in El Methni and Stupfler (2017, 2018)), Lp−quantiles (introduced in Chen (1996), and studied recently in Daouia et al (2018b, 2019) and extremiles (introduced and studied in Daouia et al (2018a))
Summary
Introduction and motivationHeavy-tailed random variables appear in numerous fields of statistical applications of extreme value analysis, such as insurance and finance (see e.g. p.9 of Embrechts et al (1997)), geoscience (see Section 1.3.5 of Beirlant et al (2004)) and analysis of teletraffic data (see Section 8 of Resnick (2007)). Recent studies geared towards insurance and finance have been advocating for the use and estimation of extreme versions of alternatives to quantiles, such as Conditional Tail Moments (El Methni et al 2014), Wang distortion risk measures (introduced in Wang (1996), and studied recently in El Methni and Stupfler (2017, 2018)), Lp−quantiles (introduced in Chen (1996), and studied recently in Daouia et al (2018b, 2019) and extremiles (introduced and studied in Daouia et al (2018a)) All these quantities, can be shown to have the heavy-tailed behaviour displayed by tail quantiles, and as such, their estimators at extreme levels can be constructed via straightforward adaptations of Weissman’s method by relying on tail index estimators as well. Tail index estimation is a central question in the statistical analysis of heavy-tailed distributions
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call