Open Access Published by De Gruyter Open Access

Volume 8 Issue 1

January 2020

Issue of Dependence Modeling

Contents
Journal Overview

Regular articles

Open Access March 25, 2020

Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property

Giovanna Nappo, Fabio Spizzichino

Page range: 1-33

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Abstract

We first review an approach that had been developed in the past years to introduce concepts of “bivariate ageing” for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing. A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. “Archimedean” models have an important role in the present framework. In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of “Pseudo-Archimedean” models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD.

Open Access March 31, 2020

Modelling with star-shaped distributions

Eckhard Liebscher, Wolf-Dieter Richter

Page range: 45-69

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Abstract

We prove and describe in great detail a general method for constructing a wide range of multivariate probability density functions. We introduce probabilistic models for a large variety of clouds of multivariate data points. In the present paper, the focus is on star-shaped distributions of an arbitrary dimension, where in case of spherical distributions dependence is modeled by a non-Gaussian density generating function.

Open Access April 3, 2020

Checkerboard copula defined by sums of random variables

Viktor Kuzmenko, Romel Salam, Stan Uryasev

Page range: 70-92

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Abstract

We consider the problem of finding checkerboard copulas for modeling multivariate distributions. A checkerboard copula is a distribution with a corresponding density defined almost everywhere by a step function on an m -uniform subdivision of the unit hyper-cube. We develop optimization procedures for finding copulas defined by multiply-stochastic matrices matching available information. Two types of information are used for building copulas: 1) Spearman Rho rank correlation coefficients; 2) Empirical distributions of sums of random variables combined with empirical marginal probability distributions. To construct checkerboard copulas we solved optimization problems. The first problem maximizes entropy with constraints on Spearman Rho coefficients. The second problem minimizes some error function to match available data. We conducted a case study illustrating the application of the developed methodology using property and casualty insurance data. The optimization problems were numerically solved with the AORDA Portfolio Safeguard (PSG) package, which has precoded entropy and error functions. Case study data, codes, and results are posted at the web.

Open Access July 4, 2020

The deFinetti representation of generalised Marshall–Olkin sequences

Henrik Sloot

Page range: 107-118

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Abstract

We show that each infinite exchangeable sequence τ 1 , τ 2 , . . . of random variables of the generalised Marshall–Olkin kind can be uniquely linked to an additive subordinator via its deFinetti representation. This is useful for simulation, model estimation, and model building.

Open Access July 4, 2020

Bayesian estimation of generalized partition of unity copulas

Andreas Masuhr, Mark Trede

Page range: 119-131

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Abstract

This paper proposes a Bayesian estimation algorithm to estimate Generalized Partition of Unity Copulas (GPUC), a class of nonparametric copulas recently introduced by [18]. The first approach is a random walk Metropolis-Hastings (RW-MH) algorithm, the second one is a random blocking random walk Metropolis-Hastings algorithm (RBRW-MH). Both approaches are Markov chain Monte Carlo methods and can cope with ˛at priors. We carry out simulation studies to determine and compare the efficiency of the algorithms. We present an empirical illustration where GPUCs are used to nonparametrically describe the dependence of exchange rate changes of the crypto-currencies Bitcoin and Ethereum.

Open Access July 27, 2020

Bivariate box plots based on quantile regression curves

Jorge Navarro

Page range: 132-156

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Abstract

In this paper, we propose a procedure to build bivariate box plots (BBP). We first obtain the theoretical BBP for a random vector ( X , Y ). They are based on the univariate box plot of X and the conditional quantile curves of Y | X . They can be computed from the copula of ( X , Y ) and the marginal distributions. The main advantage of these BBP is that the coverage probabilities of the regions are distribution-free. So they can be selected by the users with the desired probabilities and they can be used to perform fit tests. Three reasonable options are proposed. They are illustrated with two examples from a normal model and an exponential model with a Clayton copula. Moreover, several methods to estimate the theoretical BBP are discussed. The main ones are based on linear and non-linear quantile regression. The others are based on empirical estimators and parametric and non-parametric (kernel) copula estimations. All of them can be used to get empirical BBP. Some extensions for the multivariate case are proposed as well.

Open Access July 27, 2020

Bayesian credibility premium with GB2 copulas

Himchan Jeong, Emiliano A. Valdez

Page range: 157-171

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Abstract

For observations over a period of time, Bayesian credibility premium may be used to predict the value of a response variable for a subject, given previously observed values. In this article, we formulate Bayesian credibility premium under a change of probability measure within the copula framework. Such reformulation is demonstrated using the multivariate generalized beta of the second kind (GB2) distribution. Within this family of GB2 copulas, we are able to derive explicit form of Bayesian credibility premium. Numerical illustrations show the application of these estimators in determining experience-rated insurance premium. We consider generalized Pareto as a special case.

Open Access September 11, 2020

Optimizing effective numbers of tests by vine copula modeling

Nico Steffen, Thorsten Dickhaus

Page range: 172-185

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Abstract

In the multiple testing context, we utilize vine copulae for optimizing the effective number of tests. It is well known that for the calibration of multiple tests for control of the family-wise error rate the dependencies between the marginal tests are of utmost importance. It has been shown in previous work, that positive dependencies between the marginal tests can be exploited in order to derive a relaxed Šidák-type multiplicity correction. This correction can conveniently be expressed by calculating the corresponding „effective number of tests“ for a given (global) significance level. This methodology can also be applied to blocks of test statistics so that the effective number of tests can be calculated by the sum of the effective numbers of tests for each block. In the present work, we demonstrate how the power of the multiple test can be optimized by taking blocks with high inner-block dependencies. The determination of those blocks will be performed by means of an estimated vine copula model. An algorithm is presented which uses the information of the estimated vine copula to make a data-driven choice of appropriate blocks in terms of (estimated) dependencies. Numerical experiments demonstrate the usefulness of the proposed approach.

Open Access October 1, 2020

Lorenz-generated bivariate Archimedean copulas

Andrea Fontanari, Pasquale Cirillo, Cornelis W. Oosterlee

Page range: 186-209

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Abstract

A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these “Lorenz copulas”, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An “alchemy” of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.

Open Access October 1, 2020

The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay

Jan-Frederik Mai

Page range: 210-220

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Abstract

We derive a sufficient condition on the symmetric norm ||·|| such that the probability distribution associated with the density function f ( x ) ∝exp(−λ ||x||) is conditionally independent and identically distributed in the sense of de Finetti’s seminal theorem. The criterion is mild enough to comprise the ℓ p -norms as special cases, in which f is shown to correspond to a polynomially tilted stable mixture of products of transformed Gamma densities. In another special case of interest f equals the density of a time-homogeneous load sharing model, popular in reliability theory, whose motivation is a priori unrelated to the concept of conditional independence. The de Finetti structure reveals a surprising link between time-homogeneous load sharing models and the concept of Lévy subordinators.

Open Access October 1, 2020

Nonparametric relative recursive regression

Yousri Slaoui, Salah Khardani

Page range: 221-238

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Abstract

In this paper, we propose the problem of estimating a regression function recursively based on the minimization of the Mean Squared Relative Error ( MSRE ), where outlier data are present and the response variable of the model is positive. We construct an alternative estimation of the regression function using a stochastic approximation method. The Bias, variance, and Mean Integrated Squared Error ( MISE ) are computed explicitly. The asymptotic normality of the proposed estimator is also proved. Moreover, we conduct a simulation to compare the performance of our proposed estimators with that of the two classical kernel regression estimators and then through a real Malaria dataset.

Open Access October 1, 2020

Dependence uncertainty bounds for the energy score and the multivariate Gini mean difference

Carole Bernard, Alfred Müller

Page range: 239-253

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Abstract

The energy distance and energy scores became important tools in multivariate statistics and multivariate probabilistic forecasting in recent years. They are both based on the expected distance of two independent samples. In this paper we study dependence uncertainty bounds for these quantities under the assumption that we know the marginals but do not know the dependence structure. We find some interesting sharp analytic bounds, where one of them is obtained for an unusual spherically symmetric copula. These results should help to better understand the sensitivity of these measures to misspecifications in the copula.

Open Access October 5, 2020

Quadratic transformation of multivariate aggregation functions

Prakassawat Boonmee, Santi Tasena

Page range: 254-261

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Abstract

In this work, we prove that quadratic transformations of aggregation functions must come from quadratic aggregation functions. We also show that this is different from quadratic transformations of (multivariate) semi-copulas and quasi-copulas. In the latter case, those two classes are actually the same and consists of convex combinations of the identity map and another fixed quadratic transformation. In other words, it is a convex set with two extreme points. This result is different from the bivariate case in which the two classes are different and both are convex with four extreme points.

Open Access October 27, 2020

Detecting and modeling critical dependence structures between random inputs of computer models

Nazih Benoumechiara, Nicolas Bousquet, Bertrand Michel, Philippe Saint-Pierre

Page range: 263-297

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Abstract

Uncertain information on input parameters of computer models is usually modeled by considering these parameters as random, and described by marginal distributions and a dependence structure of these variables. In numerous real-world applications, while information is mainly provided by marginal distributions, typically from samples, little is really known on the dependence structure itself. Faced with this problem of incomplete or missing information, risk studies that make use of these computer models are often conducted by considering independence of input variables, at the risk of including irrelevant situations. This approach is especially used when reliability functions are considered as black-box models. Such analyses remain weakened in absence of in-depth model exploration, at the possible price of a strong risk misestimation. Considering the frequent case where the reliability output is a quantile, this article provides a methodology to improve risk assessment, by exploring a set of pessimistic dependencies using a copula-based strategy. In dimension greater than two, a greedy algorithm is provided to build input regular vine copulas reaching a minimum quantile to which a reliability admissible limit value can be compared, by selecting pairwise components of sensitive influence on the result. The strategy is tested over toy models and a real industrial case-study. The results highlight that current approaches can provide non-conservative results.

Open Access November 22, 2020

State dependent correlations in the Vasicek default model

A. Metzler

Page range: 298-329

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Abstract

This paper incorporates state dependent correlations (those that vary systematically with the state of the economy) into the Vasicek default model. Other approaches to randomizing correlation in the Vasicek model have either assumed that correlation is independent of the systematic risk factor (zero state dependence) or is an explicit function of the systematic risk factor (perfect state dependence). By contrast, our approach allows for an arbitrary degree of state dependence and includes both zero and perfect state dependence as special cases. This is accomplished by expressing the factor loading as a function of an auxiliary (Gaussian) variable that is correlated with the systematic risk factor. Using Federal Reserve data on delinquency rates we use maximum likelihood to estimate the parameters of the model, and find the empirical degree of state dependence to be quite high (but generally not perfect). We also find that randomizing correlation, without allowing for state dependence, does not improve the empirical performance of the Vasicek model.

Open Access December 16, 2020

A new extreme value copula and new families of univariate distributions based on Freund’s exponential model

Sándor Guzmics, Georg Ch. Pflug

Page range: 330-360

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Abstract

The use of the exponential distribution and its multivariate generalizations is extremely popular in lifetime modeling. Freund’s bivariate exponential model (1961) is based on the idea that the remaining lifetime of any entity in a bivariate system is shortened when the other entity defaults. Such a model can be quite useful for studying systemic risk, for instance in financial systems. Guzmics and Pflug (2019) revisited Freund’s model, deriving the corresponding bivariate copula and examined some characteristics of it; furthermore, we opened the door for a multivariate setting. Now we present further investigations in the bivariate model: we compute the tail dependence coefficients, we examine the marginal and joint distributions of the componentwise maxima, which leads to an extreme value copula, which – to the best of our knowledge – has not been investigated in the literature yet. The original bivariate model of Freund has been extended to more variables by several authors. We also turn to the multivariate setting, and our focus is different from that of the previous generalizations, and therefore it is novel: examining the distribution of the sum and of the average of the lifetime variables (provided that the shock parameters are all the same) leads to new families of univariate distributions, which we call Exponential Gamma Mixture Type I and Type II (EGM) distributions. We present their basic properties, we provide asymptotics for them, and finally we also provide the limiting distribution for the EGM Type II distribution.

Open Access December 16, 2020

Multivariate medial correlation with applications

Helena Ferreira, Marta Ferreira

Page range: 361-372

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Abstract

We define a multivariate medial correlation coefficient that extends the probabilistic interpretation and properties of Blomqvist’s β coefficient, incorporates multivariate marginal dependencies and it preserves a partial ordering stronger than concordance relation. We illustrate the results in some models and provide an application on real datasets.

Open Access December 16, 2020

Two symmetric and computationally efficient Gini correlations

Courtney Vanderford, Yongli Sang, Xin Dang

Page range: 373-395

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Abstract

Standard Gini correlation plays an important role in measuring the dependence between random variables with heavy-tailed distributions. It is based on the covariance between one variable and the rank of the other. Hence for each pair of random variables, there are two Gini correlations and they are not equal in general, which brings a substantial difficulty in interpretation. Recently, Sang et al (2016) proposed a symmetric Gini correlation based on the joint spatial rank function with a computation cost of O( n 2 ) where n is the sample size. In this paper, we study two symmetric and computationally efficient Gini correlations with the computational complexity of O( n log n ). The properties of the new symmetric Gini correlations are explored. The influence function approach is utilized to study the robustness and the asymptotic behavior of these correlations. The asymptotic relative efficiencies are considered to compare several popular correlations under symmetric distributions with different tail-heaviness as well as an asymmetric log-normal distribution. Simulation and real data application are conducted to demonstrate the desirable performance of the two new symmetric Gini correlations.

Open Access December 21, 2020

On quantile based co-risk measures and their estimation

Sebastian Fuchs, Wolfgang Trutschnig

Page range: 396-416

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Abstract

Conditional Value-at-Risk (CoVaR) is defined as the Value-at-Risk of a certain risk given that the related risk equals a given threshold (CoVaR = ) or is smaller/larger than a given threshold (CoVaR < /CoVaR ≥ ). We extend the notion of Conditional Value-at-Risk to quantile based co-risk measures that are weighted mixtures of CoVaR at different levels and hence involve the stochastic dependence that occurs among the risks and that is captured by copulas. We show that every quantile based co-risk measure is a quantile based risk measure and hence fulfills all related properties. We further discuss continuity results of quantile based co-risk measures from which consistent estimators for CoVaR < and CoVaR ≥ based risk measures immediately follow when plugging in empirical copulas. Although estimating co-risk measures based on CoVaR = is a nontrivial endeavour since conditioning on events with zero probability is necessary we show that working with so-called empirical checkerboard copulas allows to construct strongly consistent estimators for CoVaR = and related co-risk measures under very mild regularity conditions. A small simulation study illustrates the performance of the obtained estimators for special classes of copulas.

Open Access December 31, 2020

Copula modeling for discrete random vectors

Gery Geenens

Page range: 417-440

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Abstract

Copulas have now become ubiquitous statistical tools for describing, analysing and modelling dependence between random variables. Sklar’s theorem, “the fundamental theorem of copulas”, makes a clear distinction between the continuous case and the discrete case, though. In particular, the copula of a discrete random vector is not fully identifiable, which causes serious inconsistencies. In spite of this, downplaying statements may be found in the related literature, where copula methods are used for modelling dependence between discrete variables. This paper calls to reconsidering the soundness of copula modelling for discrete data. It suggests a more fundamental construction which allows copula ideas to smoothly carry over to the discrete case. Actually it is an attempt at rejuvenating some century-old ideas of Udny Yule, who mentioned a similar construction a long time before copulas got in fashion.

Interview Article

Open Access March 25, 2020

The gentleman copulist

An interview with Carlo Sempi

Christian Genest, Matthias Scherer

Page range: 34-44

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Open Access April 9, 2020

Insurance applications of dependence modeling

An interview with Edward (Jed) Frees

Christian Genest, Matthias Scherer

Page range: 93-106

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Erratum

Open Access October 5, 2020

Erratum regarding “Optimizing effective numbers of tests by vine copula modeling”

Nico Steffen, Thorsten Dickhaus

Page range: 262-262

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Abstract

We correct the definition of the family-wise error rate in our previous article “Optimizing effective numbers of tests by vine copula modeling”.

About this journal

Special Issues:

Special Issue on Econometrics and Business Analytics

Dependence Modeling is an Open Access fully peer-reviewed journal aims at providing a medium for exchanging results and ideas in the area of multivariate dependence modeling.

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To submit a paper please go to the section "Submission of manuscripts" below.
Flyer and Editorial Board

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