Premium
Beyond tail median and conditional tail expectation: Extreme risk estimation using tail L p ‐optimization
Author(s) -
Gardes Laurent,
Girard Stéphane,
Stupfler Gilles
Publication year - 2020
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12433
Subject(s) - estimator , tail dependence , mathematics , extreme value theory , statistics , asymptotic distribution , normality , range (aeronautics) , event (particle physics) , econometrics , physics , materials science , multivariate statistics , composite material , quantum mechanics
The conditional tail expectation (CTE) is an indicator of tail behavior 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 (MS), although it only gives information about the frequency of a tail event. We construct a class of tail L p ‐medians encompassing the MS and CTE. For p in (1,2), a tail L p ‐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.