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Tomaž Košir

Publications and source records attributed to Tomaž Košir.

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

Dense alternating sign matrices and imprecise copulas

In this paper, we study discrete quasi-copulas and discrete imprecise copulas of minimal range, which naturally correspond to alternating sign matrices. We identify dense alternating sign matrices as a particularly well-behaved subclass. We prove that the associated discrete imprecise copulas are always coherent and hence avoid sure loss. We further show that a particular extension of the corresponding discrete quasi-copulas to the full domain preserves this structure and yields self-dual imprecise copulas. To the best of our knowledge, this provides the first general result in the literature showing how properties established for discrete imprecise copulas can be preserved when extending them to their full-domain counterpart. We conclude the paper by showing that, for general alternating sign matrices, the associated discrete imprecise copulas need not avoid sure loss, further emphasizing the special role of the subclass of dense alternating sign matrices. We establish, however, that the class of discrete quasi-copulas of minimal range is invariant under all defect transformations

math.ST↗
arXiv2026

Jordan types commuting with a hook partition

We give a complete classification of the Jordan types occurring in the nilpotent commutator of a nilpotent matrix whose Jordan type is a hook partition. As a consequence, we also show that two partitions with the same generic commuting Jordan type need not commute with each other.

math.AC↗
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↗
arXiv2024

Discrete Imprecise Copulas

In this paper, we study discrete quasi-copulas associated with imprecise copulas. We focus on discrete imprecise copulas that are in correspondence with the Alternating Sign Matrices and provide some construction techniques of dual pairs. Additionally, we show how to obtain (full-domain) self-dual imprecise copulas through patchwork techniques. We further investigate the properties of the constructed dual pairs and prove that they are coherent and avoid sure loss.

math.ST↗
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

Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models

Standard multiparameter eigenvalue problems (MEPs) are systems of $k\ge 2$ linear $k$-parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time-invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.

math.NA↗
arXiv2021

On the singular two-parameter eigenvalue problem II

In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems, which is a key relation for many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems. In 2009, Muhič and Plestenjak extended the above relation to a class of singular two-parameter eigenvalue problems with coprime characteristic polynomials and such that all finite eigenvalues are algebraically simple. They introduced a way to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems. Using new tools, in particular the stratification theory, we extend this connection to singular two-parameter eigenvalue problems with possibly multiple eigenvalues and such that characteristic polynomials can have a nontrivial common factor.

math.NA↗
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↗
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

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↗
arXiv2019

Asymmetric linkages: maxmin vs. reflected maxmin copulas

In this paper we introduce some new copulas emerging from shock models. It was shown earlier that reflected maxmin copulas (RMM for short) are not just some specific singular copulas; they contain many important absolutely continuous copulas including the negative quadrant dependent part of the Eyraud-Farlie-Gumbel-Morgenstern class. The main goal of this paper is to develop the RMM copulas with dependent endogenous shocks and give evidence that RMM copulas may exhibit some characteristics better than the original maxmin copulas (MM for short): (1) An important evidence for that is the iteration procedure of the RMM transformation which we prove to be always convergent and we give many properties of it that are useful in applications. (2) Using this result we find also the limit of the iteration procedure of the MM transformation thus answering a question proposed earlier by Durante, Omladič, Oražem, and Ružić. (3) We give the multivariate dependent RMM copula that compares to the MM version given by Durante, Omladič, Oražem, and Ružić. In all our copulas the idiosyncratic and systemic shocks are combined via asymmetric linking functions as opposed to Marshall copulas where symmetric linking functions are used.

math.ST↗
arXiv2019

Non-exchangeability of copulas arising from shock models

When choosing the right copula for our data a key point is to distinguish the family that describes it at the best. In this respect, a better choice of the copulas could be obtained through the information about the (non)symmetry of the data. Exchangeability as a probability concept (first next to independence) has been studied since 1930's, copulas have been studied since 1950's, and even the most important class of copulas from the point of view of applications, i.e. the ones arising from shock models s.a. Marshall's copulas, have been studied since 1960's. However, the point of non-exchangeability of copulas was brought up only in 2006 and has been intensively studied ever since. One of the main contributions of this paper is the maximal asymmetry function for a family of copulas. We compute this function for the major families of shock-based copulas, i.e. Marshall, maxmin and reflected maxmin (RMM for short) copulas and also for some other important families. We compute the sharp bound of asymmetry measure $μ_\infty$, the most important of the asymmetry measures, for the family of Marshall copulas and the family of maxmin copulas, which both equal to $\frac{4}{27}\ (\approx 0.148)$. One should compare this bound to the one for the class of PQD copulas to which they belong, which is $3-2\sqrt{2}\ \approx 0.172)$, and to the general bound for all copulas that is $\frac13$. Furthermore, we give the sharp bound of the same asymmetry measure for RMM copulas which is $3-2\sqrt{2}$, compared to the same bound for NQD copulas, where they belong, which is $\sqrt{5}-2\ (\approx 0.236)$. One of our main results is also the statistical interpretation of shocks in a given model at which the maximal asymmetry measure bound is attained. These interpretations for the three families studied are illustrated by examples that should be helpful to practitioners when choosing the model for their data.

math.ST↗
arXiv2018

Reflected maxmin copulas and modelling quadrant subindependence

Copula models have become popular in different applications, including modeling shocks, in view of their ability to describe better the dependence concepts in stochastic systems. The class of maxmin copulas was recently introduced by Omladič and Ružić. It extends the well known classes of Marshall-Olkin and Marshall copulas by allowing the external shocks to have different effects on the two components of the system. By a reflection (flip) in one of the variables we introduce a new class of bivariate copulas called reflected maxmin (RMM) copulas. We explore their properties and show that symmetric RMM copulas relate to general RMM copulas similarly as do semilinear copulas relate to Marshall copulas. We transfer that relation also to maxmin copulas. We also characterize possible diagonal functions of symmetric RMM copulas.

math.ST↗
arXiv2008

On pairs of commuting nilpotent matrices

Let $B$ be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition $λ$. Then it is known that its nilpotent commutator $N_B$ is an irreducible variety and that there is a unique partition $μ$ such that the intersection of the orbit of nilpotent matrices corresponding to $μ$ with $N_B$ is dense in $N_B$. We prove that map $D$ given by $D(λ)=μ$ is an idempotent map. This answers a question of Basili and Iarrobino and gives a partial answer to a question of Panyushev. In the proof, we use the fact that for a generic matrix $A \in N_B$ the algebra generated by $A$ and $B$ is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe $D(λ)$ in terms of $λ$ if $D(λ)$ has at most two parts.

math.AC↗
arXiv2007

Semitransitive and transitive subsemigroups of the inverse symmetric semigroups

We classify minimal transitive subsemigroups of the finitary inverse symmetric semigroup modulo the classification of minimal transitive subgroups of finite symmetric groups; and semitransitive subsemigroups of the finite inverse symmetric semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups.

math.GR↗