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

Empirical likelihood inference for Haezendonck-Goovaerts risk measure

“…Instead we assume F is continuous to ensure (7) holds. So conditions (C1) and (C2) appear in [21]. The first two inequalities with respect to d 1 in (C3) ensure Lemmas 2 and 3, which are standard for an empirical likelihood method.…”

“…A rigorous derivation can be found in [21] under some conditions. The above view of Haezendonck-Goovaerts risk measure motivates us to consider the following maximum empirical likelihood estimator for h q and empirical likelihood based confidence intervals.…”

“…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.…”

Asymptotic formulas for the left truncated moments of sums with consistently varying distributed increments

“…Asymptotic behavior of the left truncated moments of random sums was considered in various fields of applied probability, including risk theory and random walks [10,11,24]. In addition, quantity in ( 3) is closely related with the Haezendonck-Goovaerts risk measure (see, for instance, [15,18,28] and [29]). To get the precise asymptotic equivalence relationship, we consider r.v.s with d.f.s from class C. The main results on the asymptotics of ( 2) and ( 3) are presented in Theorems 3 and 4 below.…”

Asymptotics for Risk Measures of Extreme Risks

“…Chapter 4 is based on the working paper of Tang and Yang (2013). After the study of the HG risk measure with a power Young function in Chapter 3, it is natural to extend our study to a general Young function.…”