Beyond tail median and conditional tail expectation: extreme risk estimation using tail $L^p$−optimisation

Laurent Gardes (1) , Stéphane Girard (2, 3) , Gilles Stupfler (4)

1 IRMA - Institut de Recherche Mathématique Avancée
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
3 STATIFY - Modèles statistiques bayésiens et des valeurs extrêmes pour données structurées et de grande dimension
4 School of Mathematical Sciences [Nottingham]

The Conditional Tail Expectation is an indicator of tail behaviour that takes into account both the frequency and magnitude of a tail event. However, the asymptotic normality of its empirical estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in actuarial and financial applications. A valuable alternative is the Median Shortfall, although it only gives information about the frequency of a tail event. We construct a class of tail Lp−medians encompassing the Median Shortfall and Conditional Tail Expectation. For p in (1, 2), a tail Lp−median depends on both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaker than a finite variance. We extrapolate this estimator and another technique to extreme levels using the heavy-tailed framework. The estimators are showcased on a simulation study and on real fire insurance data.

Mots clés: semiparametric extrapolation; Lp−optimisation; heavy-tailed distribution; extreme value statistics; asymptotic normality; conditional tail expectation; median shortfall

Domaines: Statistiques [math.ST]

Laurent Gardes, Stéphane Girard, Gilles Stupfler. Beyond tail median and conditional tail expectation: extreme risk estimation using tail $L^p$−optimisation. Scandinavian Journal of Statistics, 2020, 47 (3), pp.922--949. ⟨10.1111/sjos.12433⟩. ⟨hal-01726328v4⟩