Geometric interpretation of the residual dependence coefficient

Nolde, Natalia

doi:10.1016/j.jmva.2013.08.018

Cited by 26 publications

(49 citation statements)

References 23 publications

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“…the gauge function of (X 1 , X 2 ) is obtained as ν when − log F R ∈ RV ∞ δ , using Proposition 3.1 therein. We found η X = ζ δ , with ζ = ν(1, 1) −1 , precisely as in Nolde (2014).…”

Section: Model Of Huser and Wadsworth (2018) They Consider Scale Mixtsupporting

confidence: 70%

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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Section: Model Of Huser and Wadsworth (2018) They Consider Scale Mixtsupporting

confidence: 70%

Extremal dependence of random scale constructions

2019

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“…More refined results are obtained under the further assumption of Weibull-like tails. Section 3 discusses coefficients of tail dependence and extends Theorem 2.1 in [27] on the relationship between the coefficient of intermediate tail dependence and the geometry of the limit set S. Section 4 describes the asymptotic behaviour of the number of sample points in risk regions. The Concentration Lemma (Lemma 4.2) allows us to construct distributions with unexpected second-order expansions.…”

Section: Introductionmentioning

confidence: 94%

Samples with a limit shape, multivariate extremes, and risk

Balkema

Nolde

2020

Adv. Appl. Probab.

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Large samples from a light-tailed distribution often have a well-defined shape. This paper examines the implications of the assumption that there is a limit shape. We show that the limit shape determines the upper quantiles for a large class of random variables. These variables may be described loosely as continuous homogeneous functionals of the underlying random vector. They play an important role in evaluating risk in a multivariate setting. The paper also looks at various coefficients of tail dependence and at the distribution of the scaled sample points for large samples. The paper assumes convergence in probability rather than almost sure convergence. This results in an elegant theory. In particular, there is a simple characterization of domains of attraction.

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“…In other words, (28) says that P(X i > γ, X j > γ) is regularly-varying with index 1/η. The index is called the residual tail index [15,31]. 5 When (X i , X j ) exhibit AD (AI) then we typically have η = 1 (η < 1).…”

Section: Residual Tail Indexmentioning

confidence: 99%

Efficient Simulation for Dependent Rare Events with Applications to Extremes

Andersen

Laub

Rojas-Nandayapa

2017

Methodol Comput Appl Probab

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We consider the general problem of estimating probabilities which arise as a union of dependent events. We propose a flexible series of estimators for such probabilities, and describe variance reduction schemes applied to the proposed estimators. We derive efficiency results of the estimators in rare-event settings, in particular those associated with extremes. Finally, we examine the performance of our estimators in a numerical example.

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Geometric interpretation of the residual dependence coefficient

Cited by 26 publications

References 23 publications

Extremal dependence of random scale constructions

Extremal dependence of random scale constructions

Samples with a limit shape, multivariate extremes, and risk

Efficient Simulation for Dependent Rare Events with Applications to Extremes

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