Beyond tail median and conditional tail expectation: Extreme risk estimation using tail Lp‐optimization
本文构建了一类尾部Lp中位数，它综合了尾事件的频率和大小，其经验估计量在弱于有限方差条件下渐近正态，并通过重尾框架外推至极端水平，对精算和金融应用有价值。
Abstract 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.