Estimating L‐Functionals for Heavy‐Tailed Distributions and Application

Abstract

-functionals summarize numerous statistical parameters and actuarial risk measures. Their sample estimators are linear combinations of order statistics (-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 -functionals and establish their asymptotic normality. Our results may be applied to estimate the trimmed -moments and financial risk measures for heavy-tailed distributions.

Asymptotic distribution of the quintile share ratio estimator

“…See e.g. Brazauskas et al (2008), , Necir and Meraghni (2010) for more details. The authors developed strong approximation results for the truncated integrals, which can be automatically coupled to produce bivariate asymptotic results for the pairs of such integrals and thus, in turn, via the delta-method, to obtain desired statistical inference results for ratios and other functions of the two integrals.…”

Bias-corrected estimation in distortion risk premiums for heavy-tailed losses

“…For recent literature on statistical inference for distortion premiums, we refer to Jones and Zitikis (2003), Jones and Zitikis (2007), Centeno and Andrade (2005), Zitikis (2008a, 2008b), Brazauskas et al (2008), Greselin et al (2009), Necir andMeraghni (2009), Necir and Meraghni (2010), Brahimi et al (2011), Peng et al (2012 and the references therein.…”

Bias-reduced estimation of Wang's two-sided deviation risk measure under Levy-stable regime

“…Dans ce papier, on définit un nouvel estimateur pour la TSD en utilisant les estimateursà biais rduits des quantiles extrêmes proposés par Li et al (2010). Uneétude de simulation est effedtuée dans le but de comparer, en termes de biais et d'erreur quadratique moyenne, le nouvel estimateur avec celui introduit par Necir and Meraghni (2010). …”

“…Exploiting the extreme value theory, Necir and Meraghni (2010) proposed an alternative estimation method that extends the existing results to the more significant case where variances are infinite, which is more pertinent for dangerous risks in the areas of finance and insurance. They proposed estimators that are asymptotically normal regardless of the shape of the distribution tails.…”

“…They proposed estimators that are asymptotically normal regardless of the shape of the distribution tails. Next, we present the items that are necessary to the definition of the extreme value based TSD estimator (Necir and Meraghni, 2010). Let = n and m = m n be two sequences of integers (called trimming sequences) satisfying…”