Abstract
AbstractThe 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 Lp‐medians encompassing the MS and CTE. 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.
Highlights
A precise assessment of extreme risk is a crucial question in a number of fields of statistical applications
We provide some insight into what asymptotic result on mp(α) we should aim for under condition C1(γ)
In the case p = 2, we find back the asymptotic distribution result in Theorem 1 of [19]: n(1 − αn) m2(αn) − 1 CTE(αn)
Summary
A precise assessment of extreme risk is a crucial question in a number of fields of statistical applications. Our view is rather that the use of tail Lp−medians with p ∈ (1, 2) will be beneficial for very heavy-tailed models, in which γ is higher than the finite fourth moment threshold γ = 1/4, and possibly higher than the finite variance threshold γ = 1/2 For such values of γ and with an extreme level set to be αn = 1 − 1/n (a typical choice in applications, see e.g. recently [3, 11, 12, 27]), we shall evaluate the finite-sample performance of our estimators on simulated data sets, as well as on a real set of fire insurance data featuring an estimated value of γ larger than 1/2.
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