Max-stable random sup-measures with comonotonic tail dependence

Several objects in the Extremes literature are special instances of max-stable random sup-measures. This perspective opens connections to the theory of random sets and the theory of risk measures and makes it possible to extend corresponding notions and results from the literature with streamlined proofs. In particular, it clarifies the role of Choquet random sup-measures and their stochastic dominance property. Key tools are the LePage representation of a max-stable random sup-measure and the dual representation of its tail dependence functional. Properties such as complete randomness, continuity, separability, coupling, continuous choice, invariance and transformations are also analysed.Comment: 28 pages, 1 figur

Why scoring functions cannot assess tail properties

Motivated by the growing interest in sound forecast evaluation techniques with an emphasis on distribution tails rather than average behaviour, we investigate a fundamental question arising in this context: Can statistical features of distribution tails be elicitable, i.e. be the unique minimizer of an expected score? We demonstrate that expected scores are not suitable to distinguish genuine tail properties in a very strong sense. Specifically, we introduce the class of max-functionals, which contains key characteristics from extreme value theory, for instance the extreme value index. We show that its members fail to be elicitable and that their elicitation complexity is in fact infinite under mild regularity assumptions. Further we prove that, even if the information of a max-functional is reported via the entire distribution function, a proper scoring rule cannot separate max-functional values. These findings highlight the caution needed in forecast evaluation and statistical inference if relevant information is encoded by such functionals.Comment: 18 page

A comparative tour through the simulation algorithms for max-stable processes

Being the max-analogue of $\alpha$-stable stochastic processes, max-stable processes form one of the fundamental classes of stochastic processes. With the arrival of sufficient computational capabilities, they have become a benchmark in the analysis of spatio-temporal extreme events. Simulation is often a necessary part of inference of certain characteristics, in particular for future spatial risk assessment. In this article we give an overview over existing procedures for this task, put them into perspective of one another and make comparisons with respect to their properties making use of some new theoretical results.Comment: 20 pages, 3 tables, 3 figure

Efficient simulation of Brown-Resnick processes based on variance reduction of Gaussian processes

Brown-Resnick processes are max-stable processes that are associated to Gaussian processes. Their simulation is often based on the corresponding spectral representation which is not unique. We study to what extent simulation accuracy and efficiency can be improved by minimizing the maximal variance of the underlying Gaussian process. Such a minimization is a difficult mathematical problem that also depends on the geometry of the simulation domain. We extend Matheron's (1974) seminal contribution in two aspects: (i) making his description of a minimal maximal variance explicit for convex variograms on symmetric domains and (ii) proving that the same strategy reduces the maximal variance also for a huge class of non-convex variograms representable through a Bernstein function. A simulation study confirms that our non-costly modification can lead to substantial improvements among Gaussian representations. We also compare it with three other established algorithms.Comment: 19 pages, 3 figures, 4 tables; To appear with the Applied Probability Trus

Characterization and construction of max-stable processes

Max-stable processes provide a natural framework to model spatial extremal scenarios. Appropriate summary statistics include the extremal coefficients and the (upper) tail dependence coefficients. In this thesis, the full set of extremal coefficients of a max-stable process is captured in the so-called extremal coefficient function (ECF) and the full set of upper tail dependence coefficients in the tail correlation function (TCF). Chapter 2 deals with a complete characterization of the ECF in terms of negative definiteness. For each ECF a corresponding max-stable process is constructed, which takes an exceptional role among max-stable processes with identical ECF. This leads to sharp lower bounds for the finite dimensional distributions of arbitrary max-stable processes in terms of its ECF. Chapters 3 and 4 are concerned with the class of TCFs. Chapter 3 exhibits this class as an infinite-dimensional compact convex polytope. It is shown that the set of all TCFs (of not necessarily max-stable processes) coincides with the set of TCFs stemming from max-stable processes. Chapter 4 compares the TCFs of widely used stationary max-stable processes such as Mixed Moving Maxima, Extremal Gaussian and Brown-Resnick processes. Finally, in Chapter 5, Brown-Resnick processes on the sphere and other spaces admitting a compact group action are considered and a Mixed Moving Maxima representation is derived

Stochastic ordering in multivariate extremes

The article considers the multivariate stochastic orders of upper orthants, lower orthants and positive quadrant dependence (PQD) among simple max-stable distributions and their exponent measures. It is shown for each order that it holds for the max-stable distribution if and only if it holds for the corresponding exponent measure. The finding is non-trivial for upper orthants (and hence PQD order). From dimension $d\geq 3$ these three orders are not equivalent and a variety of phenomena can occur. However, every simple max-stable distribution PQD-dominates the corresponding independent model and is PQD-dominated by the fully dependent model. Among parametric models the asymmetric Dirichlet family and the H\"usler-Reiss family turn out to be PQD-ordered according to the natural order within their parameter spaces. For the H\"usler-Reiss family this holds true even for the supermodular order.Comment: 30 pages, 8 figure

Systematic co-occurrence of tail correlation functions among max-stable processes

The tail correlation function (TCF) is one of the most popular bivariate extremal dependence measures that has entered the literature under various names. We study to what extent the TCF can distinguish between different classes of well-known max-stable processes and identify essentially different processes sharing the same TCF.Comment: 31 pages, 4 Tables, 5 Figure

The realization problem for tail correlation functions

For a stochastic process $\{X_t\}_{t \in T}$ with identical one-dimensional margins and upper endpoint $\tau_{\text{up}}$ its tail correlation function (TCF) is defined through $\chi^{(X)}(s,t) = \lim_{\tau \to \tau_{\text{up}}} P(X_s > \tau \,\mid\, X_t > \tau )$. It is a popular bivariate summary measure that has been frequently used in the literature in order to assess tail dependence. In this article, we study its realization problem. We show that the set of all TCFs on $T \times T$ coincides with the set of TCFs stemming from a subclass of max-stable processes and can be completely characterized by a system of affine inequalities. Basic closure properties of the set of TCFs and regularity implications of the continuity of $\chi$ are derived. If $T$ is finite, the set of TCFs on $T \times T$ forms a convex polytope of $\lvert T \rvert \times \lvert T \rvert$ matrices. Several general results reveal its complex geometric structure. Up to $\lvert T \rvert = 6$ a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as $\lvert T \rvert\geq 3$ grows.Comment: 42 pages, 7 Table