arXiv subjects

Klaus Herrmann

Publications and source records attributed to Klaus Herrmann.

6 recordsLinked to original sources

arXiv2026

Morillas-type transformations of copulas and stable tail dependence functions

A stochastic representation and sampling algorithm for Morillas-type copula-to-copula transformations and related distortions of multivariate distribution functions is derived, resulting as a byproduct in a novel sampling scheme for Archimedean and Archimax copulas. This closes a methodological gap and facilitates simulation-based applications of distorted copulas. For stable tail dependence functions (stdfs), a Morillas-type distortion framework is introduced, where monomial distortions with exponents below 1 are shown to preserve stdfs via a domain-restricted Pexider equation analysis. This characterization is leveraged to identify distortions preserving extreme value copulas, and convex combinations of distorted stdfs are proposed to increase flexibility in extremal dependence modeling. The impact of distortions on maximum domain of attraction limits is also analyzed. Explicit limiting EVC distortions are identified under non-restrictive regular variation assumptions. Examples of absolutely monotone distortions allowing to fine-tune the extreme value behavior after distortion, but also of non-regularly varying 2-absolutely monotone distortions are given.

stat.ME↗

arXiv2024

Generalized extremiles and risk measures of distorted random variables

Quantiles, expectiles and extremiles can be seen as concepts defined via an optimization problem, where this optimization problem is driven by two important ingredients: the loss function as well as a distributional weight function. This leads to the formulation of a general class of functionals that contains next to the above concepts many interesting quantities, including also a subclass of distortion risks. The focus of the paper is on developing estimators for such functionals and to establish asymptotic consistency and asymptotic normality of these estimators. The advantage of the general framework is that it allows application to a very broad range of concepts, providing as such estimation tools and tools for statistical inference (for example for construction of confidence intervals) for all involved concepts. After developing the theory for the general functional we apply it to various settings, illustrating the broad applicability. In a real data example the developed tools are used in an analysis of natural disasters.

stat.ME↗

arXiv2024

Limiting Behavior of Maxima under Dependence

Weak convergence of maxima of dependent sequences of identically distributed continuous random variables is studied under normalizing sequences arising as subsequences of the normalizing sequences from an associated iid sequence. This general framework allows one to derive several generalizations of the well-known Fisher-Tippett-Gnedenko theorem under conditions on the univariate marginal distribution and the dependence structure of the sequence. The limiting distributions are shown to be compositions of a generalized extreme value distribution and a distortion function which reflects the limiting behavior of the diagonal of the underlying copula. Uniform convergence rates for the weak convergence to the limiting distribution are also derived. Examples covering well-known dependence structures are provided. Several existing results, e.g. for exchangeable sequences or stationary time series, are embedded in the proposed framework.

math.PR↗

arXiv2023

Index-mixed copulas

The class of index-mixed copulas is introduced and its properties are investigated. Index-mixed copulas are constructed from given base copulas and a random index vector, and show a rather remarkable degree of analytical tractability. The analytical form of the copula and, if it exists, its density are derived. As the construction is based on a stochastic representation, sampling algorithms can be given. Properties investigated include bivariate and trivariate margins, mixtures of index-mixed copulas, symmetries such as radial symmetry and exchangeability, tail dependence, measures of concordance such as Blomqvist's beta, Spearman's rho or Kendall's tau and concordance orderings. Examples and illustrations are provided, and applications to the distribution of sums of dependent random variables as well as the stress testing of general dependence structures are given. A particularly interesting feature of index-mixed copulas is that they allow one to provide a revealing interpretation of the well-known family of Eyraud-Farlie-Gumbel-Morgenstern (EFGM) copulas. Through the lens of index-mixing, one can explain why EFGM copulas can only model a limited range of concordance and are tail independent, for example. Index-mixed copulas do not suffer from such restrictions while remaining analytically tractable.

stat.ME↗

arXiv2021

Smooth bootstrapping of copula functionals

The smooth bootstrap for estimating copula functionals in small samples is investigated. It can be used both to gauge the distribution of the estimator in question and to augment the data. Issues arising from kernel density and distribution estimation in the copula domain are addressed, such as how to avoid the bounded domain, which bandwidth matrix to choose, and how the smoothing can be carried out. Furthermore, we investigate how the smooth bootstrap impacts the underlying dependence structure or the functionals in question and under which conditions it does not. We provide specific examples and simulations that highlight advantages and caveats of the approach.

stat.CO↗

arXiv2017

Multivariate Geometric Expectiles

A generalization of expectiles for d-dimensional multivariate distribution functions is introduced. The resulting geometric expectiles are unique solutions to a convex risk minimization problem and are given by d-dimensional vectors. They are well behaved under common data transformations and the corresponding sample version is shown to be a consistent estimator. We exemplify their usage as risk measures in a number of multivariate settings, highlighting the influence of varying margins and dependence structures.

q-fin.RM↗