Estimating 𝐿-Functionals for Heavy-Tailed Distributions and Application

Abstract

𝐿-functionals summarize numerous statistical parameters and actuarial risk measures. Their sample estimators are linear combinations of order statistics (𝐿-statistics). There exists a class of heavy-tailed distributions for which the asymptotic normality of these estimators cannot be obtained by classical results. In this paper we propose, by means of extreme value theory, alternative estimators for 𝐿-functionals and establish their asymptotic normality. Our results may be applied to estimate the trimmed 𝐿-moments and financial risk measures for heavy-tailed distributions.