arXiv subjects

Wolfgang Trutschnig

Publications and source records attributed to Wolfgang Trutschnig.

At least 19 recordsLinked to original sources

arXiv2026

How far are $d$-dimensional copulas with uniform $(d-1)$-marginals from (total) independence?

We consider the family ${\mathcal{C}}_d^{Π_{d-1}}$ of all $d$-dimensional copulas whose $(d-1)$-dimensional marginals are all equal to the $(d-1)$-dimensional product copula $Π_{d-1}$ and tackle the natural question, `how far away' from the $d$-dimensional product copula $Π_d$ elements in ${\mathcal{C}}_d^{Π_{d-1}}$ can be. We provide definitive answers for both, the uniform metric $d_\infty$ as well as the stronger, conditioning-based metric $D_1$. The established results clearly indicate that the family ${\mathcal{C}}_d^{Π_{d-1}}$ is larger than one might expect.

math.ST↗

arXiv2026

Identifiability, Convergence and Nonparametric Estimation of Bivariate Archimax Copulas

Considering that the family of bivariate Archimax copulas contains both the Archimedean and the extreme-value class, Archimax copulas constitute a flexible family allowing to model extreme and moderate levels of dependence. Despite their appeal, no fully nonparametric, consistent estimator that is itself an element of the Archimax family $\mathcal{C}_{am}$ has been established yet, mainly because Archimedean generators and Pickands dependence functions alone do not identify Archimax copulas. We resolve this identifiability issue by working with transformed generators and transformed Pickands dependence functions, and show that these functions do identify the Archimax copula uniquely. Building upon this result, we prove that uniform convergence of Archimax copulas is equivalent to uniform convergence of the corresponding transformed generators and Pickands dependence functions. Moreover, as for Archimedean and extreme-value copulas, uniform convergence in $\mathcal{C}_{am}$ is equivalent to weak convergence of almost all conditional distributions. Exploiting these equivalences, we construct two nonparametric estimators for Archimax copulas (a Pickands and a CFG type estimator, both elements of $\mathcal{C}_{am}$) and show that they are strongly consistent under mild regularity conditions. As a further consequence of the aforementioned weak conditional convergence, we obtain strongly consistent plug-in estimators for measures of directed dependence such as Chatterjee's $ξ$ and Trutschnig's $ζ_1$. A large-scale simulation study shows that the proposed CFG type estimator outperforms both the standard empirical copula estimator and the Pickands type estimator; an application to precipitation data from Bregenz and Dornbirn (Austria) illustrates the practical use of our estimators on real data.

math.ST↗

arXiv2026

On the asymptotic average diameter of blocks of uniformly distributed sequences and related results

This paper was triggered by recent results on the maximal `average distance between consecutive points' of uniformly distributed sequences (u.f.d.s.). Here we address a generalized version of this question, consider pairwise maximal/minimal/total distances in blocks/segments of $ d \geq 2 $ consecutive points of u.f.d.s., and derive sharp upper bounds for all three aggregations. Our main idea of proof consists in, firstly, adding degrees of freedom, secondly, translating the resulting problem to a solvable optimization problem over the compact family of $ d $-stochastic measures, and, thirdly, showing that the obtained bounds are also sharp bounds for the original problem.

math.NT↗

arXiv2026

On d-stochastic measures with fractal support and uniform (d-1)-marginals, and related results

The family $\mathcal{P}_{d}^{λ_{d-1}}$ of all probability measures on $[0,1]^d$ whose $(d-1)$-dimensional marginals are all equal to the Lebesgue measure $λ_{d-1}$ on $[0,1]^{d-1}$ contains remarkably pathological elements: Working with Iterated Function Systems with Probabi\-lities (IFSPs) we construct measures $μ\in \mathcal{P}_{d}^{λ_{d-1}}$ of the following two types: (i) $μ$ has self-similar fractal support; (ii) $μ$ has self-similar support and models the situation of complete/functional dependence in each direction.As our main results concerning type (i) we prove, firstly, that for every $d\geq 3$ the set $\mathcal{D}_d$ of Hausdorff dimensions of the supports of elements in $\mathcal{P}_{d}^{λ_{d-1}}$ is dense in $[d-1,d]$; and, secondly, that the subset of elements in $\mathcal{P}_{d}^{λ_{d-1}}$ having fractal support is dense in $\mathcal{P}_{d}^{λ_{d-1}}$ with respect to the Wasserstein metric. Moreover, we show the existence of an element in $\mathcal{P}_{3}^{λ_{2}}$ of type (ii) whose support is a Sierpinski tetrahedron and study some generalizations.

math.PR↗

arXiv2026

Estimating Conditional Distributions via Sklar's Theorem and Empirical Checkerboard Approximations, with Consequences to Nonparametric Regression

We tackle the natural question of whether it is possible to estimate conditional distributions via Sklar's theorem by separately estimating the conditional distributions of the underlying copula and the marginals. Working with so-called empirical checkerboard/Bernstein approximations with suitably chosen resolution/degree, we first show that uniform weak convergence to the true underlying copula can be established under very mild regularity assumptions. Building upon these results and plugging in the univariate empirical marginal distribution functions we then provide an affirmative answer to the afore-mentioned question and prove strong consistency of the resulting estimators for the conditional distributions. Moreover, we show that aggregating our estimators allows to construct consistent nonparametric estimators for the mean, the quantile, and the expectile regression function, and beyond. Some simulations illustrating the performance of the estimators and a real data example complement the established theoretical results.

math.ST↗

arXiv2025

Mean and quantile regression in the copula setting: properties, sharp bounds and a note on estimation

Driven by the interest on how uniformity of marginal distributions propa\-gates to properties of regression functions, in this contribution we tackle the following questions: Given a $(d-1)$-dimensional random vector $\textbf{X}$ and a random variable $Y$ such that all univariate marginals of $(\textbf{X},Y)$ are uniformly distributed on $[0,1]$, how large can the average absolute deviation of the mean and the quantile regression function of $Y$ given $\textbf{X}$ from the value $\frac{1}{2}$ be, and how much mass may sets with large deviation have? We answer these questions by deriving sharp inequalities, both in the mean as well as in the quantile setting, and sketch some cautionary consequences to nowadays quite popular pair copula constructions involving the so-called simplifying assumption. Rounding off our results, working with the so-called empirical checkerboard estimator in the bivariate setting, we show strong consistency for both regression types and illustrate the speed of convergence in terms of a simulation study.

math.ST↗

arXiv2024

On differentiability and mass distributions of topologically typical multivariate Archimedean copulas

Copulas, in particular Archimedean copulas are commonly viewed as analytically nice and regular objects. Motivated by a recently established result sta\-ting that the first partial derivatives of bivariate copulas can exhibit surprisingly pathological behavior, we focus on the class of $d$-dimensional Archimedean copulas denoted by $\mathcal{C}_{ar}^d$ and show that partial derivatives of order $(d-1)$ can be sur\-pri\-singly irregular as well. In fact, we prove the existence of Archimedean copulas $C \in \mathcal{C}_{ar}^d$ whose $(d-1)$-st order partial derivatives are pathological in the sense that for almost every $\mathbf{x} \in [0,1]^{d-1}$ the derivative $\partial_1...\partial_{d-1}C(\mathbf{x},y)$ does not exist on a dense set of $y \in (0,1)$. \\ Since the existence of mixed partial derivatives of order $(d-1)$ of a copula $C$ is closely related to the existence of a discrete component, we also study mass distributions of Archimedean copulas. Building upon the interplay between Archimedean copulas and so-called Williamson measures we show that absolute continuity, discreteness and singularity of the Williamson measure propagates to the associated Archimedean copula and vice versa. Moreover, we prove the fact that the sub-family of $\mathcal{C}_{ar}^d$ consisting of copulas whose absolutely continuous, discrete and singular component have full support is dense in $\mathcal{C}_{ar}^d$. \\ Finally, viewing $\mathcal{C}_{ar}^d$ in the light of Baire categories, we show that, in contrast to the space of bivariate copulas, a topologically typical $d$-dimensional Archimedean copula $C$ is not absolutely continuous but has degenerated discrete component, implying that pathological elements are rare in $\mathcal{C}_{ar}^d$.

math.PR↗

arXiv2024

On bivariate lower semilinear copulas and the star product

We revisit the family $\mathcal{C}^{LSL}$ of all bivariate lower semilinear (LSL) copulas first introduced by Durante et al. in 2008 and, using the characterization of LSL copulas in terms of diagonals with specific properties, derive several novel and partially unexpected results. In particular we prove that the star product (also known as Markov product) $S_{δ_1}*S_{δ_2}$ of two LSL copulas $S_{δ_1},S_{δ_2}$ is again a LSL copula, i.e., that the family $\mathcal{C}^{LSL}$ is closed with respect to the star product. Moreover, we show that translating the star product to the class of corresponding diagonals $\mathcal{D}^{LSL}$ allows to determine the limit of the sequence $S_δ, S_δ*S_δ, S_δ*S_δ*S_δ,\ldots$ for every diagonal $δ\in \mathcal{D}^{LSL}$. In fact, for every LSL copula $S_δ$ the sequence $(S_δ^{*n})_{n \in \mathbb{N}}$ converges to some LSL copula $S_{\overlineδ}$, the limit $S_{\overlineδ}$ is idempotent, and the class of all idempotent LSL copulas allows for a simple characterization. Complementing these results we then focus on concordance of LSL copulas. After deriving simple formulas for Kendall's $τ$ and Spearman's $ρ$ we study the exact region $Ω^{LSL}$ determined by these two concordance measures of all elements in $\mathcal{C}^{LSL}$, derive a sharp lower bound and finally show that $Ω^{LSL}$ is convex and compact.

math.ST↗

arXiv2024

On differentiability and mass distributions of typical bivariate copulas

Despite the fact that copulas are commonly considered as analytically smooth/regular objects, derivatives of copulas have to be handled with care. Triggered by a recently published result characterizing multivariate copulas via $(d-1)$-increasingness of their partial derivative we study the bivariate setting in detail and show that the set of non-differentiability points of a copula may be quite large. We first construct examples of copulas $C$ whose first partial derivative $\partial_1C(x,y)$ is pathological in the sense that for almost every $x \in (0,1)$ it does not exist on a dense subset of $y \in (0,1)$, and then show that the family of these copulas is dense. Since in commonly considered subfamilies more regularity might be typical, we then focus on bivariate Extreme Value copulas (EVC) and show that a topologically typical EVC is not absolutely continuous but has degenerated discrete component, implying that in this class typically $\partial_1C(x,y)$ exists in full $(0,1)^2$. Considering that regularity of copulas is closely related to their mass distributions we then study mass distributions of topologically typical copulas and prove the surprising fact that topologically typical bivariate copulas are mutually completely dependent with full support. Furthermore, we use the characterization of EVCs in terms of their associated Pickands dependence measures $\vartheta$ on $[0,1]$, show that regularity of $\vartheta$ carries over to the corresponding EVC and prove that the subfamily of all EVCs whose absolutely continuous, discrete and singular component has full support is dense in the class of all EVCs.

math.ST↗

arXiv2024

Revisiting the region determined by Spearman's $ρ$ and Spearman's footrule $ϕ$

Kokol and Stopar ($2023$) recently studied the exact region $Ω_{ϕ,ρ}$ determined by Spearman's footrule $ϕ$ and Spearman's $ρ$ and derived a sharp lower, as well as a non-sharp upper bound for $ρ$ given $ϕ$. Considering that the proofs for establishing these inequalities are novel and interesting, but technically quite involved we here provide alternative simpler proofs mainly building upon shuffles, symmetry, denseness and mass shifting. As a by-product of these proofs we derive several additional results on shuffle rearrangements and the interplay between diagonal copulas and shuffles which are of independent interest. Moreover we finally show that we can get closer to the (non-sharp) upper bound than established in the literature so far.

math.ST↗

arXiv2023

Quantifying and estimating dependence via sensitivity of conditional distributions

Recently established, directed dependence measures for pairs $(X,Y)$ of random variables build upon the natural idea of comparing the conditional distributions of $Y$ given $X=x$ with the marginal distribution of $Y$. They assign pairs $(X,Y)$ values in $[0,1]$, the value is $0$ if and only if $X,Y$ are independent, and it is $1$ exclusively for $Y$ being a function of $X$. Here we show that comparing randomly drawn conditional distributions with each other instead or, equivalently, analyzing how sensitive the conditional distribution of $Y$ given $X=x$ is on $x$, opens the door to constructing novel families of dependence measures $Λ_φ$ induced by general convex functions $φ: \mathbb{R} \rightarrow \mathbb{R}$, containing, e.g., Chatterjee's coefficient of correlation as special case. After establishing additional useful properties of $Λ_φ$ we focus on continuous $(X,Y)$, translate $Λ_φ$ to the copula setting, consider the $L^p$-version and establish an estimator which is strongly consistent in full generality. A real data example and a simulation study illustrate the chosen approach and the performance of the estimator. Complementing the afore-mentioned results, we show how a slight modification of the construction underlying $Λ_φ$ can be used to define new measures of explainability generalizing the fraction of explained variance.

math.ST↗

arXiv2023

A link between Kendall's tau, the length measure and the surface of bivariate copulas, and a consequence to copulas with self-similar support

Working with shuffles we establish a close link between Kendall's tau, the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well-known that Spearman's rho of a bivariate copula A is a rescaled version of the volume of the area under the graph of A, in this contribution we show that the other famous concordance measure, Kendall's tau, allows for a simple geometric interpretation as well - it is inextricably linked to the surface area of A.

math.ST↗

arXiv2022

On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform

Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class $\mathcal{C}_{ar}^d$ of all $d$-dimensional Archimedean copulas with $d \geq 3$ and proves the afore-mentioned implication with respect to conditioning on the first $d-1$ coordinates. Several proper\-ties equivalent to pointwise convergence in $\mathcal{C}_{ar}^d$ are established and - as by-product of working with conditional distributions (Markov kernels) - alternative simple proofs for the well-known formulas for the level set masses $μ_C(L_t)$ and the Kendall distribution function $F_K^d$ as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators $ψ$ of $d$-dimensional Archimedean copulas from the perspective of their so-called Williamson measures $γ$ on $(0,\infty)$ is then shown to allow not only to derive surprisingly simple expressions for $μ_C(L_t)$ and $F_K^d$ in terms of $γ$ and to characterize pointwise convergence in $\mathcal{C}_{ar}^d$ by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of $γ$ directly carry over to the corresponding copula $C_γ\in \mathcal{C}_{ar}^d$. These results are finally used to prove the fact that the family of all absolutely continuous and the family of all singular $d$-dimensional copulas is dense in $\mathcal{C}_{ar}^d$ and to underline that despite of their simple algebraic structure Archimedean copulas may exhibit surprisingly singular behavior in the sense of irregularity of their conditional distribution functions.

math.ST↗

arXiv2022

On a multivariate copula-based dependence measure and its estimation

Working with so-called linkages allows to define a copula-based, $[0,1]$-valued multivariate dependence measure $ζ^1(\boldsymbol{X},Y)$ quantifying the scale-invariant extent of dependence of a random variable $Y$ on a $d$-dimensional random vector $\boldsymbol{X}=(X_1,\ldots,X_d)$ which exhibits various good and natural properties. In particular, $ζ^1(\boldsymbol{X},Y)=0$ if and only if $\boldsymbol{X}$ and $Y$ are independent, $ζ^1(\boldsymbol{X},Y)$ is maximal exclusively if $Y$ is a function of $\boldsymbol{X}$, and ignoring one or several coordinates of $\boldsymbol{X}$ can not increase the resulting dependence value. After introducing and analyzing the metric $D_1$ underlying the construction of the dependence measure and deriving examples showing how much information can be lost by only considering all pairwise dependence values $ζ^1(X_1,Y),\ldots,ζ^1(X_d,Y)$ we derive a so-called checkerboard estimator for $ζ^1(\boldsymbol{X},Y)$ and show that it is strongly consistent in full generality, i.e., without any smoothness restrictions on the underlying copula. Some simulations illustrating the small sample performance of the estimator complement the established theoretical results.

math.ST↗

arXiv2022

Race Driver Evaluation at a Driving Simulator using a physical Model and a Machine Learning Approach

Professional race drivers are still superior to automated systems at controlling a vehicle at its dynamic limit. Gaining insight into race drivers' vehicle handling process might lead to further development in the areas of automated driving systems. We present a method to study and evaluate race drivers on a driver-in-the-loop simulator by analysing tire grip potential exploitation. Given initial data from a simulator run, two optimiser based on physical models maximise the horizontal vehicle acceleration or the tire forces, respectively. An overall performance score, a vehicle-trajectory score and a handling score are introduced to evaluate drivers. Our method is thereby completely track independent and can be used from one single corner up to a large data set. We apply the proposed method to a motorsport data set containing over 1200 laps from seven professional race drivers and two amateur drivers whose lap times are 10-20% slower. The difference to the professional drivers comes mainly from their inferior handling skills and not their choice of driving line. A downside of the presented method for certain applications is an extensive computation time. Therefore, we propose a Long-short-term memory (LSTM) neural network to estimate the driver evaluation scores. We show that the neural network is accurate and robust with a root-mean-square error between 2-5% and can replace the optimisation based method. The time for processing the data set considered in this work is reduced from 68 hours to 12 seconds, making the neural network suitable for real-time application.

cs.LG↗

arXiv2021

On distributions with fixed marginals maximizing the joint or the prior default probability, estimation, and related results

We study the problem of maximizing the probability that (i) an electric component or financial institution $X$ does not default before another component or institution $Y$ and (ii) that $X$ and $Y$ default jointly within the class of all random variables $X,Y$ with given univariate continuous distribution functions $F$ and $G$, respectively, and show that the maximization problems correspond to finding copulas maximizing the mass of the endograph $Γ^\leq(T)$ and the graph $Γ(T)$ of $T=G \circ F^-$, respectively. After providing simple, copula-based proofs for the existence of copulas attaining the two maxima $\overline{m}_T$ and $\overline{w}_T$ we generalize the obtained results to the case of general (not necessarily monotonic) transformations $T:[0,1] \rightarrow [0,1]$ and derive simple and easily calculable formulas for $\overline{m}_T$ and $\overline{w}_T$ involving the distribution function $F_T$ of $T$ (interpreted as random variable on $[0,1]$). The latter are then used to charac\-terize all non-decreasing transformations $T:[0,1] \rightarrow [0,1]$ for which $\overline{m}_T$ and $\overline{w}_T$ coincide. A strongly consistent estimator for the maximum probability that $X$ does not default before $Y$ is derived and proven to be asymptotically normal under very mild regularity conditions. Several examples and graphics illustrate the main results and falsify some seemingly natural conjectures.

math.PR↗

arXiv2021

How simplifying and flexible is the simplifying assumption in pair-copula constructions -- analytic answers in dimension three and a glimpse beyond

Motivated by the increasing popularity and the seemingly broad applicability of pair-copula constructions underlined by numerous publications in the last decade, in this contribution we tackle the unavoidable question on how flexible and simplifying the commonly used `simplifying assumption' is from an analytic perspective and provide answers to two related open questions posed by Nagler and Czado in 2016. Aiming at a simplest possible setup for deriving the main results we first focus on the three-dimensional setting. We prove that the family of simplified copulas is flexible in the sense that it is dense in the set of all three-dimensional co\-pulas with respect to the uniform metric $d_\infty$ - considering stronger notions of convergence like the one induced by the metric $D_1$, by weak conditional convergence, by total variation, or by Kullback-Leibler divergence, however, the family even turn out to be nowhere dense and hence insufficient for any kind of flexible approximation. Furthermore, returning to $d_\infty$ we show that the partial vine copula is never the optimal simplified copula approximation of a given, non-simplified copula $C$, and derive examples illustrating that the corresponding approximation error can be strikingly large and extend to more than 28\% of the diameter of the metric space. Moreover, the mapping $ψ$ assigning each three-dimensional copula its unique partial vine copula turns out to be discontinuous with respect to $d_\infty$ (but continuous with respect to $D_1$ and to weak conditional convergence), implying a surprising sensitivity of partial vine copula approximations. The afore-mentioned main results concerning $d_\infty$ are then extended to the general multivariate setting.

math.ST↗

arXiv2020

On weak conditional convergence of bivariate Archimedean and Extreme Value copulas, and consequences to nonparametric estimation

Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance - in fact, given samples of a copula $C$ the corresponding empirical copulas do not converge weakly conditional to $C$ with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and weak conditional convergence can even be proved to be equivalent. Moreover, it can be shown that every copula $C$ is the weak conditional limit of a sequence of checkerboard copulas. After proving these three main results and pointing out some consequences we sketch some implications for two recently introduced dependence measures and for the nonparametric estimation of Archimedean and Extreme Value copulas.

math.ST↗