Extremal dependence of random scale constructions

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.

Linking representations for multivariate extremes via a limit set

“…The process V E is assumed to possess hidden regular variation, with residual tail dependence coefficient satisfying η V ps 1 , s 2 q ă 1 for all s 1 ‰ s 2 . The resulting process X Ẽ is asymptotically independent for γ P p0, 1s and asymptotically dependent for γ ą 1; see also Engelke et al (2019) for related results.…”

“…In some of the examples below, it is convenient to establish a limit set and its gauge function using this observation rather than transforming to exactly exponential margins. Models with convenient dependence properties are often constructed through judicious combinations of random vectors with known dependence structures; see, for example, Engelke et al (2019) for a detailed study of so-called random scale or random location constructions. In Section 4.3, we use our results to elucidate the shape of the limit set when independent exponential-tailed variables are mixed additively.…”

Hierarchical Transformed Scale Mixtures for Flexible Modeling of Spatial Extremes on Datasets With Many Locations

“…Loosely speaking, heavier tailed R, impacting simultaneously the whole domain S, induces asymptotic dependence in X * , whereas lighter tailed R induces asymptotic independence. Engelke et al (2019) provide a fuller description of how extremal dependence of X * relates to the relative marginal tail heaviness of R and g(Z). Huser et al (2017) also used an identity link function, but placed few assumptions on the random scale, and provided more general results on the joint tail decay rates of the mixture processes.…”

Modeling spatial extremes using normal mean-variance mixtures

“…A random vector X ∈ R d is called a scale mixture if it has the representation X = RW , where R is a non-negative univariate random variable and W ∈ R d is a random vector. Scale mixtures provide a flexible family of distributions that can capture both asymptotic dependence and independence depending on the specification of the tail of R and W ; see Huser et al (2017), Huser & Wadsworth (2019), Engelke et al (2019). In practice W is often taken as a Gaussian random vector and in this case X is termed a Gaussian scale mixture.…”