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Volume 9 Issue 1

Issue of Dependence Modeling

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Journal Overview

Regular articles

Open Access March 1, 2021

Generalized Bernoulli process with long-range dependence and fractional binomial distribution

Jeonghwa Lee

Page range: 1-12

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Abstract

Bernoulli process is a finite or infinite sequence of independent binary variables, X i , i = 1, 2, · · ·, whose outcome is either 1 or 0 with probability P ( X i = 1) = p , P ( X i = 0) = 1 – p , for a fixed constant p ∈ (0, 1). We will relax the independence condition of Bernoulli variables, and develop a generalized Bernoulli process that is stationary and has auto-covariance function that obeys power law with exponent 2 H – 2, H ∈ (0, 1). Generalized Bernoulli process encompasses various forms of binary sequence from an independent binary sequence to a binary sequence that has long-range dependence. Fractional binomial random variable is defined as the sum of n consecutive variables in a generalized Bernoulli process, of particular interest is when its variance is proportional to n 2 H , if H ∈ (1/2, 1).

Open Access March 30, 2021

Polynomial bivariate copulas of degree five: characterization and some particular inequalities

Adam Šeliga, Manuel Kauers, Susanne Saminger-Platz, Radko Mesiar, Anna Kolesárová, Erich Peter Klement

Page range: 13-42

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Abstract

Bivariate polynomial copulas of degree 5 (containing the family of Eyraud-Farlie-Gumbel-Morgenstern copulas) are in a one-to-one correspondence to certain real parameter triplets ( a , b , c ), i.e., to some set of polynomials in two variables of degree 1: p ( x , y ) = ax + by + c . The set of the parameters yielding a copula is characterized and visualized in detail. Polynomial copulas of degree 5 satisfying particular (in)equalities (symmetry, Schur concavity, positive and negative quadrant dependence, ultramodularity) are discussed and characterized. Then it is shown that for polynomial copulas of degree 5 the values of several dependence parameters (including Spearman’s rho, Kendall’s tau, Blomqvist’s beta, and Gini’s gamma) lie in exactly the same intervals as for the Eyraud-Farlie-Gumbel-Morgenstern copulas. Finally we prove that these dependence parameters attain all possible values in ]−1, 1[ if polynomial copulas of arbitrary degree are considered.

Open Access May 24, 2021

Multivariate radial symmetry of copula functions: finite sample comparison in the i.i.d case

Monica Billio, Lorenzo Frattarolo, Dominique Guégan

Page range: 43-61

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Abstract

Given a d -dimensional random vector X = ( X 1 , . . ., X d ), if the standard uniform vector U obtained by the component-wise probability integral transform (PIT) of X has the same distribution of its point reflection through the center of the unit hypercube, then X is said to have copula radial symmetry. We generalize to higher dimensions the bivariate test introduced in [11], using three different possibilities for estimating copula derivatives under the null. In a comprehensive simulation study, we assess the finite-sample properties of the resulting tests, comparing them with the finite-sample performance of the multivariate competitors introduced in [17] and [1].

Open Access June 4, 2021

Explaining predictive models using Shapley values and non-parametric vine copulas

Kjersti Aas, Thomas Nagler, Martin Jullum, Anders Løland

Page range: 62-81

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Abstract

In this paper the goal is to explain predictions from complex machine learning models. One method that has become very popular during the last few years is Shapley values. The original development of Shapley values for prediction explanation relied on the assumption that the features being described were independent. If the features in reality are dependent this may lead to incorrect explanations. Hence, there have recently been attempts of appropriately modelling/estimating the dependence between the features. Although the previously proposed methods clearly outperform the traditional approach assuming independence, they have their weaknesses. In this paper we propose two new approaches for modelling the dependence between the features. Both approaches are based on vine copulas, which are flexible tools for modelling multivariate non-Gaussian distributions able to characterise a wide range of complex dependencies. The performance of the proposed methods is evaluated on simulated data sets and a real data set. The experiments demonstrate that the vine copula approaches give more accurate approximations to the true Shapley values than their competitors.

Open Access July 5, 2021

Study of partial and average conditional Kendall’s tau

Irène Gijbels, Margot Matterne

Page range: 82-120

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Abstract

When the interest is in studying conditional dependencies, and more precisely the strength of conditional dependencies, some kind of averaging over the conditioning random vector may be needed. Examples of average measures that can serve in this context are the average conditional Kendall’s tau and partial Kendall’s tau. It is known that these measures differ in general. Some statistical tests are based on these average measures, and a better knowledge of them is of importance. The aim of this paper is to provide a quantitative study of the possible differences of these two average measures, and to establish su˚cient conditions under which they coincide. Both measures are studied in two fairly general settings. In each setting theoretical results are established as well as several illustrative examples given.

Open Access July 23, 2021

Detecting departures from meta-ellipticity for multivariate stationary time series

Axel Bücher, Miriam Jaser, Aleksey Min

Page range: 121-140

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Abstract

A test for detecting departures from meta-ellipticity for multivariate stationary time series is proposed. The large sample behavior of the test statistic is shown to depend in a complicated way on the underlying copula as well as on the serial dependence. Valid asymptotic critical values are obtained by a bootstrap device based on subsampling. The finite-sample performance of the test is investigated in a large-scale simulation study, and the theoretical results are illustrated by a case study involving financial log returns.

Open Access July 31, 2021

Generalized Bernoulli process: simulation, estimation, and application

Jeonghwa Lee

Page range: 141-155

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Abstract

A generalized Bernoulli process (GBP) is a stationary process consisting of binary variables that can capture long-memory property. In this paper, we propose a simulation method for a sample path of GBP and an estimation method for the parameters in GBP. Method of moments estimation and maximum likelihood estimation are compared through empirical results from simulation. Application of GBP in earthquake data during the years of 1800-2020 in the region of conterminous U.S. is provided.

Open Access August 18, 2021

Asymptotic normality of the relative error regression function estimator for censored and time series data

Feriel Bouhadjera, Elias Ould Saïd

Page range: 156-178

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Abstract

Consider a survival time study, where a sequence of possibly censored failure times is observed with d-dimensional covariate The main goal of this article is to establish the asymptotic normality of the kernel estimator of the relative error regression function when the data exhibit some kind of dependency. The asymptotic variance is explicitly given. Some simulations are drawn to lend further support to our theoretical result and illustrate the good accuracy of the studied method. Furthermore, a real data example is treated to show the good quality of the prediction and that the true data are well inside in the confidence intervals.

Open Access September 25, 2021

Hoeffding–Sobol decomposition of homogeneous co-survival functions: from Choquet representation to extreme value theory application

Cécile Mercadier, Paul Ressel

Page range: 179-198

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Abstract

The paper investigates the Hoeffding–Sobol decomposition of homogeneous co-survival functions. For this class, the Choquet representation is transferred to the terms of the functional decomposition, and in addition to their individual variances, or to the superset combinations of those. The domain of integration in the resulting formulae is reduced in comparison with the already known expressions. When the function under study is the stable tail dependence function of a random vector, ranking these superset indices corresponds to clustering the components of the random vector with respect to their asymptotic dependence. Their Choquet representation is the main ingredient in deriving a sharp upper bound for the quantities involved in the tail dependograph, a graph in extreme value theory that summarizes asymptotic dependence.

Open Access December 13, 2021

Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

Guillaume Boglioni Beaulieu, Pierre Lafaye de Micheaux, Frédéric Ouimet

Page range: 424-438

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Abstract

We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of F ). This allows us to illustrate the extent of the ‘failure’ of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer K , new K -tuplewise independent sequences that are not mutually independent. For K [four.tf], it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

Open Access December 30, 2021

Detection of arbitrage opportunities in multi-asset derivatives markets

Antonis Papapantoleon, Paulo Yanez Sarmiento

Page range: 439-459

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Abstract

We are interested in the existence of equivalent martingale measures and the detection of arbitrage opportunities in markets where several multi-asset derivatives are traded simultaneously. More specifically, we consider a financial market with multiple traded assets whose marginal risk-neutral distributions are known, and assume that several derivatives written on these assets are traded simultaneously. In this setting, there is a bijection between the existence of an equivalent martingale measure and the existence of a copula that couples these marginals. Using this bijection and recent results on improved Fréchet–Hoeffding bounds in the presence of additional information on functionals of a copula by [18], we can extend the results of [33] on the detection of arbitrage opportunities to the general multi-dimensional case. More specifically, we derive sufficient conditions for the absence of arbitrage and formulate an optimization problem for the detection of a possible arbitrage opportunity. This problem can be solved efficiently using numerical optimization routines. The most interesting practical outcome is the following: we can construct a financial market where each multi-asset derivative is traded within its own no-arbitrage interval, and yet when considered together an arbitrage opportunity may arise.

Special Issue in Memory of Abe Sklar

Open Access September 30, 2021

Special Issue on copulas in memory of Abe Sklar (1925-2020)

Giovanni Puccetti

Page range: 199-199

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Open Access September 30, 2021

A tribute to Abe Sklar

Christian Genest

Page range: 200-224

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Open Access October 6, 2021

On partially Schur-constant models and their associated copulas

Claude Lefèvre

Page range: 225-242

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Abstract

Schur-constant vectors are used to model duration phenomena in various areas of economics and statistics. They form a particular class of exchangeable vectors and, as such, rely on a strong property of symmetry. To broaden the field of applications, partially Schur-constant vectors are introduced which correspond to partially exchangeable vectors. First, their copulas of survival, said to be partially Archimedean, are explicitly obtained and analyzed. Next, much attention is devoted to the construction of different partially Schur-constant models with two groups of exchangeable variables. Finally, partial Schur-constancy is briefly extended to the modeling of nested and multi-level dependencies.

Open Access October 11, 2021

On copulas of self-similar Ito processes

Piotr Jaworski, Marcin Krzywda

Page range: 243-266

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Abstract

We characterize the cumulative distribution functions and copulas of two-dimensional self-similar Ito processes, with randomly correlated Wiener margins, as solutions of certain elliptic partial differential equations.

Open Access October 18, 2021

Sklar’s theorem, copula products, and ordering results in factor models

Jonathan Ansari, Ludger Rüschendorf

Page range: 267-306

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Abstract

We consider a completely specified factor model for a risk vector X = ( X 1 , . . ., X d ), where the joint distributions of the components of X with a risk factor Z and the conditional distributions of X given Z are specified. We extend the notion of *-product of d -copulas as introduced for d = 2 and continuous factor distribution in Darsow et al. [6] and Durante et al. [8] to the multivariate and discontinuous case. We give a Sklar-type representation theorem for factor models showing that these *-products determine the copula of a completely specified factor model. We investigate in detail approximation, transformation, and ordering properties of *-products and, based on them, derive general orthant ordering results for completely specified factor models in dependence on their specifications. The paper generalizes previously known ordering results for the worst case partially specified risk factor models to some general classes of positive or negative dependent risk factor models. In particular, it develops some tools to derive sharp worst case dependence bounds in subclasses of completely specified factor models.

Open Access October 18, 2021

On convergence of associative copulas and related results

Thimo M. Kasper, Sebastian Fuchs, Wolfgang Trutschnig

Page range: 307-326

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Abstract

Triggered by a recent article establishing the surprising result that within the class of bivariate Archimedean copulas 𝒞 ar different notions of convergence - standard uniform convergence, convergence with respect to the metric D 1 , and so-called weak conditional convergence - coincide, in the current contribution we tackle the natural question, whether the obtained equivalence also holds in the larger class of associative copulas 𝒞 a . Building upon the fact that each associative copula can be expressed as (finite or countably infinite) ordinal sum of Archimedean copulas and the minimum copula M we show that standard uniform convergence and convergence with respect to D 1 are indeed equivalent in 𝒞 a . It remains an open question whether the equivalence also extends to weak conditional convergence. As by-products of some preliminary steps needed for the proof of the main result we answer two conjectures going back to Durante et al. and show that, in the language of Baire categories, when working with D 1 a typical associative copula is Archimedean and a typical Archimedean copula is strict.

Open Access October 26, 2021

Generating unfavourable VaR scenarios under Solvency II with patchwork copulas

Dietmar Pfeifer, Olena Ragulina

Page range: 327-346

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Abstract

The central idea of the paper is to present a general simple patchwork construction principle for multivariate copulas that create unfavourable VaR (i.e. Value at Risk) scenarios while maintaining given marginal distributions. This is of particular interest for the construction of Internal Models in the insurance industry under Solvency II in the European Union. Besides this, the Delegated Regulation by the European Commission requires all insurance companies under supervision to consider different risk scenarios in their risk management system for the company’s own risk assessment. Since it is unreasonable to assume that the potential worst case scenario will materialize in the company, we think that a modelling of various unfavourable scenarios as described in this paper is likewise appropriate. Our explicit copula approach can be considered as a special case of ordinal sums, which in two dimensions even leads to the technically worst VaR scenario.

Open Access October 28, 2021

New results on perturbation-based copulas

Susanne Saminger-Platz, Anna Kolesárová, Adam Šeliga, Radko Mesiar, Erich Peter Klement

Page range: 347-373

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Abstract

A prominent example of a perturbation of the bivariate product copula (which characterizes stochastic independence) is the parametric family of Eyraud-Farlie-Gumbel-Morgenstern copulas which allows small dependencies to be modeled. We introduce and discuss several perturbations, some of them perturbing the product copula, while others perturb general copulas. A particularly interesting case is the perturbation of the product based on two functions in one variable where we highlight several special phenomena, e.g., extremal perturbed copulas. The constructions of the perturbations in this paper include three different types of ordinal sums as well as flippings and the survival copula. Some particular relationships to the Markov product and several dependence parameters for the perturbed copulas considered here are also given.

Open Access October 30, 2021

On a general class of gamma based copulas

Barry C. Arnold, Matthew Arvanitis

Page range: 374-384

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Abstract

A large family of copulas with gamma components is examined, and interesting submodels are defined and analyzed. Parameter estimation is demonstrated for some of these submodels. A brief discussion of higher-dimensional versions is included.

Open Access December 4, 2021

Dispersive order comparisons on extreme order statistics from homogeneous dependent random vectors

Mhamed Mesfioui, Julien Trufin

Page range: 385-393

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Abstract

In this paper, we investigate sufficient conditions for preservation property of the dispersive order for the smallest and largest order statistics of homogeneous dependent random vectors. Moreover, we establish sufficient conditions for ordering with the dispersive order the largest order statistics from dependent homogeneous samples of different sizes.

Open Access December 4, 2021

Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

Rachele Foschi, Giovanna Nappo, Fabio L. Spizzichino

Page range: 394-423

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Abstract

As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T 1 , ..., Tr . In any case, we assume that T 1 , ..., Tr are identically distributed, with a common survival function ̄G and their survival copula is denoted by K . The diagonal sections of K , along with ̄G , are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T 1 , ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results.

About this journal

Special Issue:

The journal Dependence Modeling aims at providing a medium for exchanging results and ideas in the area of multivariate dependence modeling.
It is an open access fully peer-reviewed journal providing the readers with free, instant, and permanent access to all content worldwide.
Dependence Modeling is listed by Web of Science (Emerging Sources Citation Index), Scopus, MathSciNet and Zentralblatt Math.

To submit a paper please go to the section ‘’Submission of manuscripts’’ below.
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The journal presents different types of articles:

"Research Articles" on fundamental theoretical aspects, as well as on significant applications in science, engineering, economics, finance, insurance and other fields.
"Review Articles" which present the existing literature on the specific topic from new perspectives.
"Interview articles" limited to two papers per year, covering interviews with milestone personalities in the field of Dependence Modeling.

The journal topics include (but are not limited to):　

Copula methods
Multivariate distributions
Estimation and goodness-of-fit tests
Measures of association
Quantitative risk management
Risk measures and stochastic orders
Time series
Environmental sciences
Computational methods and software
Extreme-value theory
Limit laws
Mass Transportations