Abstract
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.
Highlights
A rich variety of bivariate dependence models have a pseudo-polar representation (X1, X2) = R(W1, W2), R > 0, independent of (W1, W2) ∈ W ⊆ R2, (1)where we term R the radial variable, assumed to have a non-degenerate distribution, and (W1, W2) the angular variables
These restrictions mean that λ ≤ 0 corresponds to asymptotic independence; the residual tail dependence coefficient ηX is as given in Proposition 2 for λ = 0 with ζ = τ (1/2) = (1, 1) −∗ 1, and Proposition 3 for λ < 0
∈ RV∞ δ, using precisely as in Various papers focus on extremal dependence arising from certain types of polar representation, but from a conditional extremes perspective (Heffernan and Tawn 2004; Heffernan and Resnick 2007). This is different to our focus; here we examine the extremal dependence as both variables grow at the same rate
Summary
Where W is two-dimensional, the possible constructions stemming from Eq 1 form an especially large class, since (W1, W2) can itself have any copula In this case, we focus on how the multiplication by R changes the extremal dependence of (W1, W2), summarized by the coefficients (χW , ηW ), to obtain the extremal dependence of the modified vector (X1, X2) in terms of its coefficients (χX, ηX). The Gumbel limit in particular attracts distributions with highly diverse tail behavior such as finite upper bounds or heavy tails Overall, this classification is too coarse for our requirements, and it excludes important classes such as superheavy-tailed distributions defined through the property of heavy-tailed log-transformed random variables.
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