An exceptional max-stable process fully parameterized by its extremal coefficients

Abstract

The extremal coefficient function (ECF) of a max-stable process X on some index set T assigns to each finite subset A ⊂ T the effective number of independent random variables among the collection {Xt}t∈A. We introduce the class of Tawn-Molchanov processes that is in a 1:1 correspondence with the class of ECFs, thus also proving a complete characterization of the ECF in terms of negative definiteness. The corresponding Tawn-Molchanov process turns out to be exceptional among all max-stable processes sharing the same ECF in that its dependency set is maximal w.r.t. inclusion. This entails sharp lower bounds for the finite dimensional distributions of arbitrary max-stable processes in terms of its ECF. A spectral representation of the Tawn-Molchanov process and stochastic continuity are discussed. We also show how to build new valid ECFs from given ECFs by means of Bernstein functions.

Distributionally robust inference for extreme Value-at-Risk

“…Such types of spectral measures correspond to the Tawn-Molchanov max-stable model [28]. This is an interesting finding since, as shown in the last reference, the Tawn-Molchanov max-stable models are maximal elements with respect to the lower orthant stochastic order, for the set of all max-stable distributions sharing a fixed set of extremal coefficients.…”

“…We show first that the minimization problem (L ρ ) reduces to a standard linear program. Interestingly, val(L ρ ) is attained by spectral measures corresponding to the celebrated Tawn-Molchanov max-stable models [28]. This leads to efficient and exact solutions in practice for moderate number of constraints and dimensions.…”

“…A Tawn-Molchanov minimizer and a convex maximizer. Surprisingly, the infimum of ρ and in turn the lower bound on X is attained by measures with the same support as the celebrated Tawn-Molchanov models in Strokorb and Schlather [28]. This allows us to further reduce the optimization to a linear program, which can be solved exactly for moderate dimension.…”

Tail dependence of recursive max-linear models with regularly varying noise variables

“…They have been investigated, generalized, and applied to real world problems by many researchers; see e.g. Schlather and Tawn (2002), Wang and Stoev (2011), Falk et al (2015), Strokorb and Schlather (2015), Einmahl et al (2012), Cui and Zhang (2017), and Kiriliouk (2017).…”

“…The TDC, which goes back to Sibuya (1960), measures the extremal dependence between two random variables and is a simple and popular dependence measure in extreme value theory. Methods to construct multivariate max-stable distributions with given TDCs have been proposed, for example, by Schlather and Tawn (2002), Falk (2005), Falk et al (2015), and Strokorb and Schlather (2015). Somehow related we identify all RMLMs with the same given TDCs.…”