Bivariate tail estimation: dependence in asymptotic independence

Draisma, Gerrit; Drees, Holger; Ferreira, Ana; Haan, Leo de

doi:10.3150/bj/1082380219

Cited by 135 publications

(170 citation statements)

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“…de Haan and Sinha (1999), Drees and de Haan (2015), Draisma et al (2004)), the estimators (5.10) and (5.15) exploit homogeneity of a function describing tail dependence; in this case, homogeneity (3.4) of the rate function I . This offers the advantage that no explicit estimate of I is required.…”

Section: Discussionmentioning

confidence: 99%

“…Therefore, in addition, a correction of the classical estimator based on residual tail dependence (4.1) was applied cf. Draisma et al (2004).…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Later, e.g. in de Haan and Sinha (1999), Peng (1999), Drees and de Haan (2015), Draisma et al (2004) and de Haan and Ferreira (2006), convergence to the limiting distribution and its effect on bias in estimates is explicitly considered. For the marginals, additional assumptions on convergence to the limit (1.2) in these articles are identical to or stronger than those invoked for univariate quantile estimation.…”

Section: Introductionmentioning

confidence: 99%

“…To alleviate this problem, residual tail dependence (RTD), also known as hidden regular variation, was introduced as an additional regularity assumption on the tail of the multivariate survival function F c , defined by F c (x) = P (X i > x i , i = 1, .., m); e.g. Ledford and Tawn (1996, 1997, Peng (1999), Resnick (2002), Draisma et al (2004) and Heffernan and Resnick (2005). This model was recently extended in Wadsworth and Tawn (2013).…”

Section: Introductionmentioning

confidence: 99%

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Approximation and estimation of very small probabilities of multivariate extreme events

Valk

2016

Extremes

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This article discusses modelling of the tail of a multivariate distribution function by means of a large deviation principle (LDP), and its application to the estimation of the probability p n of a multivariate extreme event from a sample of n iid random vectors, with p n ∈ [n −τ 2 , n −τ 1 ] for some τ 1 > 1 and τ 2 > τ 1 . One way to view the classical tail limits is as limits of probability ratios. In contrast, the tail LDP provides asymptotic bounds or limits for log-probability ratios. After standardising the marginals to standard exponential, tail dependence is represented by a homogeneous rate function I . Furthermore, the tail LDP can be extended to represent both dependence and marginals, the latter implying marginal log-Generalised Weibull tail limits. A connection is established between the tail LDP and residual tail dependence (or hidden regular variation) and a recent extension of it. Under a smoothness assumption, they are implied by the tail LDP. Based on the tail LDP, a simple estimator for very small probabilities of extreme events is formulated. It avoids estimation of I by making use of its homogeneity. Strong consistency in the sense of convergence of log-probability ratios is proven. Simulations and an application illustrate the difference between the classical approach and the LDP-based approach.

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Section: Discussionmentioning

confidence: 99%

“…Therefore, in addition, a correction of the classical estimator based on residual tail dependence (4.1) was applied cf. Draisma et al (2004).…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximation and estimation of very small probabilities of multivariate extreme events

Valk

2016

Extremes

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“…Various cases of hidden regular variation have recently received considerable attention as part of the program to distinguish statistically asymptotic independence from dependence; see Campos et al (2005), Coles et al (1999), de Haan and de Ronde (1998), Draisma et al (2004), Heffernan (2000), Tawn (1996), (1997), Maulik and Resnick (2005), Peng (1999), Poon et al (2003), Resnick (2004), and Stȃricȃ (1999). In particular, hidden regular variation is based on Tawn's (1996), (1997) analysis of the coefficient of tail dependence.…”

Section: Introductionmentioning

confidence: 99%

Hidden regular variation and the rank transform

Heffernan¹,

Resnick²

2005

Adv. Appl. Probab.

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Random vectors in the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying with asymptotic independence. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn. We show that the rank transform that brings unequal marginals to the standard case also preserves the hidden regular variation. We discuss applications of the results to two examples, one involving flood risk and the other Internet data.

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