Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2010, Article ID 707146, 34 pages doi:10.1155/2010/707146 Research Article Estimating L-Functionals for Heavy-Tailed Distributions and Application Abdelhakim Necir and Djamel Meraghni Laboratory of Applied Mathematics, Mohamed Khider University of Biskra, 07000 Biskra, Algeria Correspondence should be addressed to Abdelhakim Necir, necirabdelhakim@yahoo.fr Received 5 October 2009; Accepted 21 January 2010 Academic Editor: Riˇ cardas Zitikis Copyright q 2010 A. Necir and D. Meraghni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 1. Introduction 1.1. L-Functionals Let X be a real random variable rv with continuous distribution function df F. The corresponding L-functionals are defined by LJ  :  1 0 J sQsds, 1.1 where Qs : inf{x ∈ R : Fx ≥ s}, 0 <s ≤ 1, is the quantile function pertaining to df F and J is a measurable function defined on 0, 1see, e.g. Serfling, 1. Several authors have used the quantity LJ  to solve some statistical problems. For example, in a work by Chernoff et al. 2 the L-functionals have a connection with optimal estimators of location and scale parameters in parametric families of distributions. Hosking 3 introduced the L-moments as a new approach of statistical inference of location, dispersion, skewness, kurtosis, and other