Abstract
We extend the statistical toolbox for inferring the extreme value behavior of stochastic processes by defining the spectrogram. It is based on a pseudo-polar representation of bivariate data and represents a collection of spectral measures that characterize the bivariate extremal dependence structure in terms of a univariate distribution. Based on threshold exceedances in the original event data, estimation of the spectrogram is flexible and efficient. The virtues of the spectrogram for exploratory studies and for parametric inference are highlighted. We propose a variance reduction technique that can be applied to the estimation of summary statistics like the extremogram. Parametric inference based on distance measures between the empirical and the model spectrogram, as for instance pairwise likelihoods or least squares distances, is developed. Simulated data illustrate gains in parametric estimation efficiency compared to a standard estimation approach. An application to precipitation data collected in the Cevennes region in France shows the practical utility of the introduced notions.
Highlights
Due to their potential for more efficient and more flexible estimation in comparison to an often used composite likelihood approach with block maxima [34]
It is constituted from the spectral measures for point pairs in space-time and provides a generalized and unified view on dependence characterizations like the extremal coefficient function [42] or the tail correlation function, a variant of the extremogram [14]
The remainder of this paper is organized as follows: At the beginning of Section 2, we first present some necessary background on extreme value theory and max-stable processes
Summary
The fundamental limit theorem of extreme value theory states that convergence of a process of rescaled pointwise maxima leads to a max-stable limit process [19]: for n independent copies {Xi(s), s ∈ D} of a process X defined on a nonempty measurable set D ⊂ Rd, we assume that max i=1,...,n an(s)−1(Xi(s). When continuity is assured on a compact domain D, each max-stable process has its Pareto equivalent and vice versa. Defines a max-stable field on D with unit Frechet margins. This construction was interpreted by Smith [44] to introduce a class of models sometimes referred to as storm profile processes: a uniform Poisson process {Si} on Rd is used to define Qi(s) = f (s, Si) for some non-negative kernel function f : Rd × Rd → R+, f (s1, s2)ds2 = 1 for all s1, which describes the deterministic shape of storms. Univariate marginal distributions Gj of a max-stable process are generalized extreme value distributions (GEV): Gj (z) = GEVξj ,νj,σj (z) = exp 1 + ξj σj−1 (z − νj ).
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