On the estimation of the variability in the distribution tail

Laurent Gardes (1) , Stéphane Girard (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 STATIFY - Modèles statistiques bayésiens et des valeurs extrêmes pour données structurées et de grande dimension

We propose a new measure of variability in the tail of a distribution by applying a Box-Cox transformation of parameter $p ≥ 0$ to the tail-Gini functional. It is shown that the so-called Box-Cox Tail Gini Variability measure is a valid variability measure whose condition of existence may be as weak as necessary thanks to the tuning parameter p. The tail behaviour of the measure is investigated under a general extreme-value condition on the distribution tail. We then show how to estimate the Box-Cox Tail Gini Variability measure within the range of the data. These methods provide us with basic estimators that are then extrapolated using the extreme-value assumption to estimate the variability in the very far tails. The finite sample behavior of the estimators is illustrated both on simulated and real data.

Mots clés: Gini functional; risk measure; variability measure; distribution tail; extreme-value theory

Domaines: Statistiques [math.ST]

Laurent Gardes, Stéphane Girard. On the estimation of the variability in the distribution tail. Test, 2021, 30, pp.884--907. ⟨10.1007/s11749-021-00754-2⟩. ⟨hal-02400320v2⟩