Second-order regular variation, convolution and the central limit theorem

Geluk, Jaap; Haan, Leo de; Resnick, Sidney I.; Stărică, Cătălin

doi:10.1016/s0304-4149(97)00042-2

Cited by 60 publications

(39 citation statements)

References 21 publications

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“…The asymptotic normality of Π ρ,n is related to that of Hill's estimator γ n which is equivalent to the second-order regular variation condition connected with (4) (see, e.g., Geluk et al (1997)). A df F fulfills the second-order regular variation condition with second-order parameter ω ≤ 0, if there exists a positive or negative function a (·) with lim t→∞ a (t) = 0, such that…”

Section: Defining the Estimatormentioning

confidence: 99%

Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts

Necir

Meraghni

2009

Insurance: Mathematics and Economics

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Section: Defining the Estimatormentioning

confidence: 99%

Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts

Necir

Meraghni

2009

Insurance: Mathematics and Economics

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“…setup under the additional assumption that k/ log log n → ∞. The asymptotic normality with a deterministic centring by 1/α requires additional assumptions on the distribution F of X and has been established in Haeusler and Teugels (1985); de Haan and ;Geluk, de Haan, S.I., and Starica (1997);Hill (2010). In this case,…”

Section: The Hill Methods For Tail Index Estimationmentioning

confidence: 99%

On tail index estimation based on multivariate data

Dematteo

Clémençon

2015

Journal of Nonparametric Statistics

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This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, i.e. of which Pareto-like marginals share the same tail index. A multivariate Central Limit Theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common tail index α, is established under mild conditions. Motivated by the statistical analysis of extremal spatial data in particular, we introduce the concept of (standard) heavy-tailed random field of tail index α and show how this limit result can be used in order to build an estimator of α with small asymptotic mean squared error, through a proper convex linear combination of the coordinates. Beyond asymptotic results, simulation experiments illustrating the relevance of the approach promoted are also presented.

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“…2RV was originally introduced to study the convergence rate of the extreme order statistics in extreme value theory (see de Ferreira 2006 andResnick 1996). In Geluk et al (1997), the equivalence of the second-order regular variation and asymptotic normality of the Hill's estimator was discussed, and it was shown that 2RV is an essential assumption in establishing asymptotic normality of Hill estimators. We refer to Degen et al (2010) for study on risk concentration based on Value-at-Risk for a portfolio and Hua and Joe (2011) for asymptotic analysis on Conditional Tail Expectation under the assumption of 2RV, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Regarding the second-order expansion of tail probabilities of S n , there is extensive literature. Among all those results, Geluk et al (1997) and made the early efforts to the case of n = 2 by dealing the special cases of identical margins and when the first-order index of regular variation equals to −1, respectively. For general n, presented a second-order result for the sum S n of n independent and identically distributed (iid) random variables with a common distribution in the Hall-Weissman class which is a special case of 2RV.…”

Section: Introductionmentioning

confidence: 99%

Second-order properties of tail probabilities of sums and randomly weighted sums

Mao

2015

Extremes

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Let X 1 , . . . , X n be independent nonnegative random variables with respective survival functions F 1 , . . . , F n , and let 1 , . . . , n be (not necessarily independent) nonnegative random variables, independent of X 1 , . . . , X n , satisfying certain moment conditions. This paper consists of two parts. In the first part, we investigate second-order expansions of P n i=1 X i > t as t → ∞ under the assumption that the F i are of second-order regular variation (2RV) with the same first-order index but with different second-order indexes. In the second part, under the assumption that the F 1 = · · · = F n have 2RV tails, second-order expansions of tail probabilities of the randomly weighted sum n i=1 i X i are studied. The closure property of 2RV under randomly weighted sum is also discussed. The main results in this paper generalize and strengthen several known results in the literature.

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Second-order regular variation, convolution and the central limit theorem

Cited by 60 publications

References 21 publications

Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts

Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts

On tail index estimation based on multivariate data

Second-order properties of tail probabilities of sums and randomly weighted sums

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