Spectral density regression for bivariate extremes

Abstract

We introduce a density regression model for the spectral density of a bivariate extreme value distribution, that allows us to assess how extremal dependence can change over a covariate. Inference is performed through a double kernel estimator, which can be seen as an extension of the Nadaraya-Watson estimator where the usual scalar responses are replaced by mean-constrained densities on the unit interval. Numerical experiments with the methods illustrate their resilience in a variety of contexts of practical interest. An extreme temperature dataset is used to illustrate our methods.

Concentration bounds for the empirical angular measure with statistical learning applications

“…It lies at the basis for a test of the hypothesis that a bivariate distribution is in the maximum domain of attraction of an extreme value distribution [22]. It serves to model the action of a covariate on the extremal dependence of a baseline distribution through a density ratio model [13,7]. The angular density is also at the basis of an estimator of bivariate tail quantile regions [21].…”

“…The combination of the marginal standardization (4) with the change-ofvariable formula in (7) implies the identities…”

“…In fact, any non-negative Borel measure Φ on S satisfying the moment constraints (8) is the angular measure of some random vector X. Indeed, from such a measure Φ on S, we can define a measure µ on E through (7) and then consider the max-stable distribution with µ as exponent measure as in [17].…”

“…M 0 , where µ is determined by the right-hand side of the previous display via the values of µ(E \ [0, x]) for x ∈ (0, ∞) d . The angular measure Φ determines the exponent measure µ uniquely via the relation (7) in the paper. In that identity, letting f be the indicator function of the set E \ [0, x] with x ∈ (0, ∞) d , it follows by a standard calculation involving Fubini's theorem that…”

Regression‐type models for extremal dependence

“…Some exceptions include de Carvalho and Davison (), who proposed a nonparametric approach, where a family of spectral densities is constructed using exponential tilting. Castro‐Camillo and de Carvalho () developed an extension of this approach based on covariate‐varying spectral densities. However, these approaches are limited to replicated one‐way ANOVA types of settings.…”