Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks

First- and Second-order Asymptotics for the Tail Distortion Risk Measure of Extreme Risks

“…From this point of view, the use of extreme value theory becomes appropriate. Similar asymptotic analysis of risk measures have recently Downloaded by [Northwestern University] at 20:11 26 December 2014 Yang been implemented by Tang and Yang (2012), Mao and Hu (2012), and Zhu and Li (2012), among others.…”

Extreme value analysis of the Haezendonck–Goovaerts risk measure with a general Young function

“…This risk measure was first introduced by Haezendonck and Goovaerts (1982) based on the Swiss premium calculation principle induced by the Orlicz norm and was revisited by Goovaerts et al (2004). Since recently, it has attracted increasing attention from researchers; see Rosazza Gianin (2008a,b, 2012), Krätschmer and Zähle (2011), Nam et al (2011), Goovaerts et al (2012), Mao and Hu (2012), Tang and Yang (2012), Cheung and Lo (2013) and Ahn and Shyamalkumar (2014), among others. As pointed out by Bellini and Rosazza Gianin (2012), the HG risk measure is a law invariant and coherent risk measure for a convex Young function ϕ(·).…”

Empirical likelihood inference for Haezendonck-Goovaerts risk measure

“…Therefore, the first equation in Lemma 1 follows from (11), (12), (13) and (16). The rest can be shown similarly.…”

“…Recently there has been an increasing interest in studying this risk measure with applications in actuarial science. For example, [10] showed that this risk measure preserves the convex order property; [3,5] provided a dual representation for this risk measure; [11] investigated a relationship between this risk measure and others; [9] obtained a lower bound for this risk measure when a sum of random variables is concerned; studies of optimal portfolio and optimal reinsurance under this risk measure are given by [4] and [22], respectively; [20], [21] derived a first order approximation for this risk measure when the underlying distribution is in the domain of attraction of an extreme value distribution, which is of importance in predicting extreme risks; a second order approximation for this risk measure is obtained by [13], which is necessary for the study of estimating this risk measure nonparametrically when the level q depends on the sample size and goes to one as the sample size tends to infinity; nonparametric estimation for this risk measure is proposed by Ahn and Shyamalkumar [1] and its asymptotic limit is derived too.…”