Extremes of randomly scaled Gumbel risks

Dȩbicki, Krzysztof; Farkas, Julia; Hashorva, Enkelejd

doi:10.1016/j.jmaa.2017.08.055

Cited by 7 publications

(4 citation statements)

References 39 publications

(44 reference statements)

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“…VR and vMR apply for all MDA distributions with arbitrary

γ \in ℝ

, not only for

γ \neq 0

as it is the case in regular variation approach. In particular, VR and vMR are often used for Gumbel MDA distributions, see Barbe & Seifert (2016), Dȩbicki et al (2018), Fougères & Soulier (2012), Hashorva (2012), Klüppelberg & Lindner (2005) and Klüppelberg & Seifert (2020) for VR, as well as Abdous et al (2008), Engelke et al (2019), Fasen et al (2006), Janßen (2010) and Seifert (2016) for vMR; this Gumbel case grants the most interesting analysis and results, as we will point out in the following sections. Using VR and vMR could be also convenient instead of applying the concept of regular variation suitable for Fréchet and Weibull MDA distributions characterized by parameter

γ \neq 0

(cf.…”

Section: Variation and Von Mises Representations Of Mda Distributionsmentioning

confidence: 99%

Characterization of valid auxiliary functions for representations of extreme value distributions and their max‐domains of attraction

Seifert

2024

Scandinavian J Statistics

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In this paper we study two important representations for extreme value distributions and their max‐domains of attraction (MDA), namely von Mises representation (vMR) and variation representation (VR), which are convenient ways to gain limit results. Both VR and vMR are defined via so‐called auxiliary functions ψ. Up to now, however, the set of valid auxiliary functions for vMR has neither been characterized completely nor separated from those for VR. We contribute to the current literature by introducing “universal” auxiliary functions which are valid for both VR and vMR representations for the entire MDA distribution families. Then we identify exactly the sets of valid auxiliary functions for both VR and vMR. Moreover, we propose a method for finding appropriate auxiliary functions with analytically simple structure and provide them for several important distributions.This article is protected by copyright. All rights reserved.

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“…VR and vMR apply for all MDA distributions with arbitrary

γ \in ℝ

, not only for

γ \neq 0

γ \neq 0

(cf.…”

Section: Variation and Von Mises Representations Of Mda Distributionsmentioning

confidence: 99%

Characterization of valid auxiliary functions for representations of extreme value distributions and their max‐domains of attraction

Seifert

2024

Scandinavian J Statistics

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“…Lemma 7 (Tail decay of products of Weibull-type variables, see Theorem 2.1(b) of Dȩbicki et al (2018)). If two independent random variables Y 1 ≥ 0 and Y 2 ≥ 0 are Weibull-tailed such that F Yj (x) ∼ r j (x) exp(−α j x βj ), x → ∞, j = 1, 2, (33) with r j ∈ RV ∞ γj .…”

Section: Proofs For Sectionmentioning

confidence: 99%

Extremal dependence of random scale constructions

2019

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A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X1, X2) = R(W1, W2), with non-degenerate R > 0 independent of (W1, W2). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W1, W2), the shape of the support of (W1, W2), and dependence between (W1, W2). When R is distinctly lighter tailed than (W1, W2), the extremal dependence of (X1, X2) is typically the same as that of (W1, W2), whereas similar or heavier tails for R compared to (W1, W2) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X1, X2) possible in such cases when (W1, W2) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.

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“…Coles et al [8]) for a pair of such portfolios. Dȩbicki et al [12] investigate the distribution of losses in the Gumbel max-domain of attraction which are scaled by random factors. However, none of these papers study consequences of risk sharing in a network or system context.…”

Section: Introductionmentioning

confidence: 99%

Financial risk measures for a network of individual agents holding portfolios of light-tailed objects

Klüppelberg

Seifert

2019

Finance Stoch

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We investigate a financial network of agents holding portfolios of independent light-tailed risky objects whose losses are asymptotically exponentially distributed with distinct tail parameters. We show that the asymptotic distributions of portfolio losses belong to the class of functional exponential mixtures. We also provide statements for Value-at-Risk and Expected Shortfall risk measures as well as for their conditional counterparts. We establish important qualitative differences in the asymptotic behavior of portfolio risks under a light tail assumption, compared to heavy tail settings, which have to be accounted for in practical risk management.

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Extremes of randomly scaled Gumbel risks

Cited by 7 publications

References 39 publications

Characterization of valid auxiliary functions for representations of extreme value distributions and their max‐domains of attraction

Characterization of valid auxiliary functions for representations of extreme value distributions and their max‐domains of attraction

Extremal dependence of random scale constructions

Financial risk measures for a network of individual agents holding portfolios of light-tailed objects

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