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

Das, Bikramjit; Mitra, Abhimanyu; Resnick, Sidney I.

doi:10.1239/aap/1363354106

Cited by 41 publications

(50 citation statements)

References 18 publications

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“…The definitions of multivariate regular variation (MRV) and hidden regular variation (HRV) are reviewed in Section 1.4 where general concepts are adapted for subcones in two dimensions. In Section 2, assuming asymptotic limit measures are specified, we adapt the standard multiplicative method for generating regularly varying models based on the generalized polar coordinate transform [10,21,26] to E = R 2 + \ {0} and E 0 = (0, ∞) 2 producing relatively tractable models.…”

Section: 2mentioning

confidence: 99%

“…Without loss of generality [21], we may assume the functions are uniformly continuous. [10,18,21] and Definition 1.1.…”

Section: Rv βmentioning

confidence: 99%

“…We review material from [10,18,21] describing the framework for the definition of MRV and HRV specialized to two dimensions. Note that the convergence concept used for defining regular variation is M-convergence which is slightly different from vague convergence traditionally used in such cases.…”

Section: Regularly Varying Distributions On Conesmentioning

confidence: 99%

“…Note that the convergence concept used for defining regular variation is M-convergence which is slightly different from vague convergence traditionally used in such cases. Reasons for preferring M-convergence are discussed in Remark 1.1 below and in [10,21].…”

Section: Regularly Varying Distributions On Conesmentioning

confidence: 99%

“…A subset C ⊂ R 2 + is a cone if it is closed under positive scalar multiplication: if x ∈ C then c x ∈ C for c > 0. A proper framework for discussing regular variation is measure convergence defined by M-convergence [10,21] on a closed cone C ⊂ R 2 + with a closed cone C 0 ⊂ C deleted. Call the deleted cone C 0 the forbidden zone.…”

Section: Regularly Varying Distributions On Conesmentioning

confidence: 99%

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Models with Hidden Regular Variation: Generation and Detection

Das

Resnick

2015

Stochastic Systems

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We review the notions of multivariate regular variation (MRV) and hidden regular variation (HRV) for distributions of random vectors and then discuss methods for generating models exhibiting both properties concentrating on the non-negative orthant in dimension two. Furthermore we suggest diagnostic techniques that detect these properties in multivariate data and indicate when models exhibiting both MRV and HRV are plausible fits for the data. We illustrate our techniques on simulated data, as well as two real Internet data sets.1. Introduction. This paper discusses methods for constructing multivariate heavy-tailed models on particular sub-cones of R 2 + = [0, ∞) 2 by adapting techniques from multivariate regular variation theory. We evaluate two different methods for generating models exhibiting multiple heavy-tailed regimes. The results obtained in the different regimes are governed by the sub-cone that serves as the state-space, the choice of scaling function, and often the interaction between different regimes. We also discuss statistical detection methods which validate that data is consistent with particular multivariate heavy-tailed models; these methods are adapted from those developed for the conditional extreme value model (CEV) [8]. We discuss multivariate regular variation (MRV) on the cones R 2 + \ {0} and (0, ∞) 2 . When regular variation exists on both cones, the regular variation on the smaller cone (0, ∞) 2 is called hidden regular variation (HRV).Data that may be modeled by distributions having heavy tails appear in many contexts, for example, hydrology [1], finance [31], insurance [14], Internet traffic [6], social networks and random graphs [3,13,28,30] and risk

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Section: 2mentioning

confidence: 99%

“…Without loss of generality [21], we may assume the functions are uniformly continuous. [10,18,21] and Definition 1.1.…”

Section: Rv βmentioning

confidence: 99%