arXiv subjects

Matjaž Omladič

Publications and source records attributed to Matjaž Omladič.

At least 19 recordsLinked to original sources

arXiv2026

The relation between non-exchangeability and radial asymmetry of copulas

Non-exchangeability and radial asymmetry have got substantial attention in the literature, but relation between the two has not been studied yet. We propose the question: if a copula has a given degree of radial asymmetry, what is the range of possible values of the measure of non-exhangeability. We determine the set of all points whose first coordinate is the radial asymmetry of a copula and the second coordinate is its non-exchangeability. We show that this set is a non-convex hexagon.

math.ST↗

arXiv2026

Maximal signed volume for (multivariate) supermodular quasi-copulas

Copulas are the primary tool for dependence modeling in statistics, and quasi-copulas are their essential companions. The latter appear, say, as infima or suprema of sets of copulas; they form a huge class and have some unpleasant properties. Their statistical interpretation is challenged by the fact that they may lead to negative volumes of some boxes. So, numerous applications call for an intermediate class, and supermodular quasi-copulas are one of them, having many useful properties. An excellent measure, Average Rectangular Volume (ARV in short), to clarify and position this class was proposed in the seminal paper by Anzilli and Durante, The average rectangular volume induced by supermodular aggregation functions, J. Math. Anal. Appl. 555 (2026) 21 pp. While supermodularity is a bivariate notion, its extension to the $d$-variate case for $d>2$ was recently emphasized in a key paper by Arias-Garcia, Mesiar, and De Baets, The unwalked path between quasi-copulas and copulas: Stepping stones in higher dimensions, Int. J. of Appr. Reasoning, 80 (2017) pp. 89-99. Here, an alternative method to ARV is presented, extendable to the multivariate case based on Maximal (in absolute value) Negative Volumes (MNV in short) on boxes, thus helping practitioners when seeking the right (quasi-)copula for their problem. Observe that these volumes on copulas are zero, while their values on quasi-copulas, depending on $d$, have been a long-standing open problem solved only recently. We present a nontrivial extension of this solution, which serves as the main goal of this paper: a measure that clarifies and positions the classes considered based on MNV.

math.ST↗

arXiv2025

A complete characterization of maximal copulas with a given track section

Bivariate copulas with prescribed diagonal section were first studied by Bertino. Their maximality was studied so far only from the point of view of upper bounds which brings quasi-copulas into the picture and limits the resulting set substantially. We propose to study maximality of these families in the order theoretic sense. A copula C with given diagonal section δ is called undominated if there is no copula C' {\neq} C with the same diagonal section δ such that C {\leq} C'. The main contribution of this paper is a new method that provides copulas of the kind. This method generates a much wider class that contains the known upper bounds as a very small subclass. There was a recent call for the study of asymmetry which is addressed by our class better than by the known ones. Corresponding quasi-copulas can be obtained from our copulas via splicing techniques. Most results are given on the level of tracks.

math.ST↗

arXiv2025

Marshall-Olkin copulas revisited

Almost seventy years old Marshall-Olkin copulas, then wider Marshall copulas, and finally even wider shock model (SM) copulas constitute a substantial part of nowadays copula theory due to numerous applications. Recently, Christian Genest with some coauthors introduced a new stochastic model for a special subclass of SM copulas which gives not only a new angle on these copulas but also widens the range of applications. In this paper we extend this type of stochastic model to all known subclasses of SM copulas. We also introduce a novel class of SM copulas and extend the new stochastic model to this class as well.

math.ST↗

arXiv2025

Extreme mass distributions for quasi-copulas

A recent survey, nicknamed "Hitchhiker's Guide", J.J. Arias-Garcıa, R. Mesiar, and B. De Baets, A hitchhiker's guide to quasi-copulas, Fuzzy Sets and Systems 393 (2020) 1-28, has raised the rating of quasi-copula problems in the dependence modeling community in spite of the lack of statistical interpretation of quasi-copulas. In our previous work (Fuzzy Sets and Systems 517 (2025) 109457), we addressed the question of extreme values of the mass distribution associated with multivariate quasi-copulas. Using a linear programming approach, we were able to solve Open Problem 5 of the "Guide" up to dimension d = 17 and disprove a recent conjecture on the solution to that problem. In this paper, we use an analytical approach to provide a complete answer to the original question.

stat.ML↗

arXiv2025

Freedom in constructing quasi-copulas vs. copulas

The main goal of this paper is to study the extent of freedom one has in constructing quasi-copulas vs. copulas. Specifically, it exhibits three construction methods for quasi-copulas based on recent developments: a representation of multivariate quasi-copulas by means of infima and suprema of copulas, an extension of a classical result on shuffles of min to the setting of quasi-copulas, and a construction method for quasi-copulas obeying a given signed mass pattern on a patch.

math.ST↗

arXiv2025

Extreme values of the mass distribution associated with $d$-quasi-copulas via linear programming

The recent survey published in Fuzzy Sets and Systems nicknamed ``Hitchhiker's Guide'' has raised the rating of quasi-copula problems in the dependence modeling community in spite of the lack of statistical interpretation of quasi-copulas. Some of the open problems listed there were solved, and some conjectured one way or the other. This paper concentrates on the Open Problem 5 of this list concerning bounds on the volume of a $d$--variate quasi-copula. We disprove a recent conjecture published in the same journal on the lower bound of this volume. We also give evidence that the problem is much more difficult than suspected and provide hints about its final solution.

math.ST↗

arXiv2023

Coherence and avoidance of sure loss for standardized functions and semicopulas

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A\leq C\leq B$ for standardized $n$-variate functions $A,B$ and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A\leq C\leq B$.

math.ST↗

arXiv2023

The solution of the Loewy-Radwan conjecture

A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of this space where the maximal dimension is attained. Extensions of this result in the direction of linear spaces of matrices with a bounded number of eigenvalues have been studied. In this paper, we answer perhaps the most general problem of the kind as proposed by Loewy and Radwan by solving their conjecture in the positive. We give the dimension of a maximal linear space of $n\times n$ matrices with no more than $k<n$ eigenvalues. We also exhibit the structure of the space where this dimension is attained.

math.RA↗

arXiv2023

On approximate and actual reducibility of matrix groups

We introduce the notions of $\varepsilon$-approximate fixed point and weak $\varepsilon$-approximate fixed point. We show that for a group of unitary matrices even the existence of a nontrivial weak $\varepsilon$-approximate fixed point for sufficiently small $\varepsilon$ gives an actual nontrivial common eigenvector. We give estimates for $\varepsilon$ in terms of the size $n$ of matrices and prove that the dependence is polynomial. Moreover, we show that the common eigenvector is polynomially close to the starting weak approximate fixed point.

math.GR↗

arXiv2023

Extending multivariate sub-quasi-copulas

In this paper, we introduce patchwork constructions for multivariate quasi-copulas. These results appear to be new since the kind of approach has been limited to either copulas or only bivariate quasi-copulas so far. It seems that the multivariate case is much more involved since we are able to prove that some of the known methods of bivariate constructions cannot be extended to higher dimensions. Our main result is to present the necessary and sufficient conditions both on the patch and the values of it for the desired multivariate quasi-copula to exist. We also give all possible solutions.

math.ST↗

arXiv2023

On approximate commutativity of spaces of matrices

The maximal dimension of commutative subspaces of $M_n(\mathbb{C})$ is known. So is the structure of such a subspace when the maximal dimension is achieved. We consider extensions of these results and ask the following natural questions: If $V$ is a subspace of $M_n(\mathbb{C})$ and $k$ is an integer less than $n$, such that for every pair $A$ and $B$ of members of $V$, the rank of the commutator $AB - BA$ is at most $k$, then how large can the dimension of $V$ be? If this maximum is achieved, can we determine the structure of $V$? We answer the first question. We also propose a conjecture on the second question which implies, in particular, that such a subspace $V$ has to be an algebra, just as in the known case of $k = 0$. We prove the proposed structure of $V$ if it is already assumed to be an algebra.

math.RA↗

arXiv2021

Spearman's footrule and Gini's gamma: Local bounds for bivariate copulas and the exact region with respect to Blomqvist's beta

Copulas are becoming an essential tool in analyzing data thus encouraging interest in related questions. In the early stage of exploratory data analysis, say, it is helpful to know local copula bounds with a fixed value of a given measure of association. These bounds have been computed for Spearman's rho, Kendall's tau, and Blomqvist's beta. The importance of another two measures of association, Spearman's footrule and Gini's gamma, has been reconfirmed recently. It is the main purpose of this paper to fill in the gap and present the mentioned local bounds for these two measures as well. It turns out that this is a quite non-trivial endeavor as the bounds are quasi-copulas that are not copulas for certain values of the two measures. We also give relations between these two measures of association and Blomqvist's beta.

math.ST↗

arXiv2020

Some multivariate imprecise shock model copulas

Bivariate imprecise copulas have recently attracted substantial attention. However, the multivariate case seems still to be a "blank slate". It is then natural that this idea be tested first on shock model induced copulas, a family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation with the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to produce their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. In addition, we unfold the theory of bivariate imprecise RMM copulas that has not yet been done before.

math.PR↗

arXiv2020

A full scale Sklar's theorem in the imprecise setting

In this paper we present a surprisingly general extension of the main result of a paper that appeared in this journal: I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48--66. The main tools we develop in order to do so are: (1) a theory on quasi-distributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) $p$-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) $p$-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladič and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) $p$-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual $σ$-additive probability space or on an additive space can be obtained as a copula of its margins meaning that its possible extraordinariness depends solely on its margins. This might indicate that copulas are a stronger probability concept than probability itself.

math.PR↗

arXiv2019

Relation between non-exchangeability and measures of concordance of copulas

An investigation is presented of how a comprehensive choice of five most important measures of concordance (namely Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta, and their weaker counterpart Spearman's footrule) relate to non-exchangeability, i.e., asymmetry on copulas. Besides these results, the method proposed also seems to be new and may serve as a raw model for exploration of the relationship between a specific property of a copula and some of its measures of dependence structure, or perhaps the relationship between various measures of dependence structure themselves.

math.ST↗

arXiv2019

Constructing copulas from shock models with imprecise distributions

The omnipotence of copulas when modeling dependence given marg\-inal distributions in a multivariate stochastic situation is assured by the Sklar's theorem. Montes et al.\ (2015) suggest the notion of what they call an \emph{imprecise copula} that brings some of its power in bivariate case to the imprecise setting. When there is imprecision about the marginals, one can model the available information by means of $p$-boxes, that are pairs of ordered distribution functions. By analogy they introduce pairs of bivariate functions satisfying certain conditions. In this paper we introduce the imprecise versions of some classes of copulas emerging from shock models that are important in applications. The so obtained pairs of functions are not only imprecise copulas but satisfy an even stronger condition. The fact that this condition really is stronger is shown in Omladič and Stopar (2019) thus raising the importance of our results. The main technical difficulty in developing our imprecise copulas lies in introducing an appropriate stochastic order on these bivariate objects.

math.PR↗

arXiv2019

Relation between Blomqvist's beta and other measures of concordance of copulas

An investigation is presented of how a comprehensive choice of four most important measures of concordance (namely Spearman's rho, Kendall's tau, Spearman's footrule, and Gini's gamma) relate to the fifth one, i.e., the Blomqvist's beta. In order to work out these results we present a novel method of estimating the values of the four measures of concordance on a family of copulas with fixed value of beta. These results are primarily aimed at the community of practitioners trying to find the right copula to be employed on their data. However, the proposed method as such may be of independent interest from theoretical point of view.

math.ST↗