Simple models for multivariate regular variation and the Hüsler–Reiß Pareto distribution

2019 ◽  
Vol 173 ◽  
pp. 525-550 ◽  
Author(s):  
Zhen Wai Olivier Ho ◽  
Clément Dombry
Keyword(s):  
Regular Variation ◽  
Pareto Distribution ◽  
Multivariate Regular Variation ◽  
Simple Models

Almost sure limit sets of random samples in ℝd

1988 ◽  
Vol 20 (3) ◽  
pp. 573-599 ◽  
Author(s):  
Richard A. Davis ◽  
Edward Mulrow ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Almost Sure Convergence ◽  
Random Sets ◽  
Regularity Conditions ◽  
Limit Set ◽  
Multivariate Regular Variation ◽  
Exponential Form ◽  
Limit Sets ◽  
Random Samples ◽  
Distribution Tail

If {Xj, } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

scholarly journals Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation

2013 ◽  
Vol 45 (01) ◽  
pp. 139-163 ◽  
Author(s):  
Bikramjit Das ◽  
Abhimanyu Mitra ◽  
Sidney Resnick
Keyword(s):  
Classical Theory ◽  
Regular Variation ◽  
Environmental Science ◽  
Asymptotic Model ◽  
Multivariate Regular Variation ◽  
Tail Risk ◽  
Risk Estimates ◽  
Hidden Regular Variation ◽  
Definition Of

Multivariate regular variation plays a role in assessing tail risk in diverse applications such as finance, telecommunications, insurance, and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to inaccurate and useless estimates of probabilities of joint tail regions. This problem can be partly ameliorated by using hidden regular variation (see Resnick (2002) and Mitra and Resnick (2011)). We offer a more flexible definition of hidden regular variation that provides improved risk estimates for a larger class of tail risk regions.

Conditional excess risk measures and multivariate regular variation

2019 ◽  
Vol 36 (1-4) ◽  
pp. 1-23
Author(s):  
Bikramjit Das ◽  
Vicky Fasen-Hartmann
Keyword(s):  
Risk Factors ◽  
Asymptotic Behavior ◽  
Systemic Risk ◽  
Regular Variation ◽  
Risk Measures ◽  
Excess Risk ◽  
Multivariate Regular Variation ◽  
Marginal Expected Shortfall ◽  
Broad Framework ◽  
Heavy Tailed

Abstract Conditional excess risk measures like Marginal Expected Shortfall and Marginal Mean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable.

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