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

Laurent Gardes 1 Stephane Girard 2 Gilles Stupfler 3
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Grenoble INP [2020-....] - Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....], Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann
Abstract : 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.
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Laurent Gardes, Stephane Girard, Gilles Stupfler. Beyond tail median and conditional tail expectation: extreme risk estimation using tail $L^p$−optimisation. Scandinavian Journal of Statistics, Wiley, 2020, 47 (3), pp.922-949. ⟨10.1111/sjos.12433⟩. ⟨hal-01726328v4⟩

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