Book of abstracts Copulas and Their Applications To commemorate the 75th birthday of Professor Roger B. Nelsen

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1 Book of abstracts Copulas and Their Applications To commemorate the 75th birthday of Professor Roger B. Nelsen July 3-5, 2017 Almería, Spain Editors: Enrique de Amo Artero, Juan Fernández Sánchez, Manuel Úbeda Flores

2 TITLE: Book of abstracts: Copulas and Their Applications. To Commemorate the 75th Birthday of Professor Roger B. Nelsen EDITORS: Enrique de Amo Artero, Juan Fernández Sánchez, Manuel Úbeda Flores. ISBN:

3 Contents Roger B. Nelsen 5 Conference Location 6 General Information 7 Program 8 s 11 Plenary Speakers 13 Invited Sessions 15 Contributed Sessions 35 List of Participants 63 3

4 Scientific Committee: Bernard De Baets (Ghent University) Hans De Meyer (Ghent University) Manuel Díaz Carrillo (Universidad de Granada) Fabrizio Durante (Università del Salento) Arturo Erdely (Universidad Nacional Autónoma de México) Piotr Jaworski (University of Warsaw) Gaspar Mayor (Universitat de les Illes Balears) Radko Mesiar (Slovak University of Technology) Johanna Nešlehová (McGill University) José Antonio Rodríguez Lallena (Universidad de Almería) Ludger Rüschendorf (Albert-Ludwig-Universität Freiburg ) Wolfgang Trutschnig (Universität Salzburg) 4

5 Roger B. Nelsen Roger Nelsen is Professor Emeritus of Mathematics at Lewis & Clark College in Portland, Oregon, USA. He studied Mathematics at DePauw University (BA, 1964) and Duke University (PhD, 1969). Roger joined the faculty at Lewis & Clark in the fall of 1969, and retired in Prior to Lewis & Clark Roger spent a year with the Biostatistics Unit of the Centre International de Recherche sur le Cancer in Lyon, France. He has had visiting appointments at the University of Massachusetts in Amherst and Mount Holyoke College in South Hadley, Massachusetts. In addition to his monograph An Introduction to Copulas, Roger has authored or co-authored 11 books published by the Mathematical Association of America. He has served on the editorial boards of two MAA journals and several of their book series. 5

6 Conference Location The conference is taking place in Aulario III (Universidad de Almerı a) Phone (Department of Mathematics): Contact phones:

7 General Information Registration Registration opens on Monday July 3 at 08:00. Location: Hall of Aulario III (building 15 in the map). Lunch and Coffee Breaks Coffee breaks and lunches are included in the conference fees. Lunches will take place in the canteen of the Central Building (building 2 in the map). Social Events Social events: On Tuesday evening we will first have a visit (approx. 45 minutes) to Los Refugios de la Guerra Civil Española ( Shelters of the Spanish Civil War ), located at Plaza Manuel Pérez García 1, in downtown. will take a short walk around the historic center and visit, if possible, the interior of the Cathedral. After the visit, we will have a social dinner in la Croquetería La Mala, located in calle Gabriel Callejón 10, in downtown. The joint cost of both events is 30 euros. Tickets for these events can be ordered during the congress in the registration table at Aulario III. Airport-Hotels-Airport Taxi: This is the easiest solution, but it cost around euros Bus: It is the worse solution because its time-table: it is very imprecise. You need to ask by there for the bus number 20. The bus goes strightly to the Centre of the town Calle Gregorio Marañón and Rambla Federico García Lorca, near from the Hotels zone. That is the last bus stop. For the back, the same bus at this bus stop also runs. To arrive to the Universidad de Almería It is 5 km form the centre of the city. The travel is lesser than half an hour (25 ) by the line coast of Almería. Buses numbers 4, 11, 12, 15 y 18 are good, with a frequence of 20. The return from the University to the city: The same buses with the same frequence. Buses Tickets: The cost is 1.05 euros per travel. You can buy it at the bus. 7

8 Program Monday Time Speaker Title of the talk 08:00-09:00 Registration 09:00-09:10 Opening 09:10-10:00 Special Talk Chair: Enrique de Amo Roger B. Nelsen My introduction to Copulas 10:00-10:20 Chair:Ludger Rüschendorf Hans De Meyer Symmetry properties of a (quasi-)copula with respect to a curve 10:20-10:40 Coffee break 10:40-12:00 Recent advances in dependence modelling and optimization Organizer & Chair: Wolfgang Trutschnig 10:40-11:00 Elena Di Bernardino Copulas and (some) multivariate risk models 11:00-11:20 Giovanni Puccetti Multivariate notions of extremal dependence and the swapping algorithm 11:20-11:40 Songkiat Sumetkijakan Non-atomic copulas 11:40-12:00 Wolfgang Trutschnig Complete dependence everywhere? 12:00-12:15 Mini-break 12:15-13:00 Measures Chair: Matthias Scherer 12:15-12:30 Maximilian Coblenz The Length of Copula Level Curves 12: Sebastian Fuchs Estimators for a Class of Bivariate Measures of Concordance for Copulas 12:45-13:00 Mathias Pohl Optimal transport in R 2 13:00-14:20 Lunch break 14:20-16:00 Recent copula constructions Organizer & Chair: Radko Mesiar 14:20-14:40 José J. Arias García Intermediate classes between quasi-copulas and copulas in higher dimensions 14:40-15:00 Tomáš Bacigál Some open problems in statistical interpretation Williamson s n-transform 15:00-15:20 Michal Dibala Discrete copulas and maximal entropy principle 15:20-15:40 L ubomíra Horanská On generalized symmetry of copulas 15:40-16:00 Matjaz Omladič Inverse-maxmin copulas and their function in modelling quadrant subindependence 16:00-16:20 Coffee break 16:20-17:50 Appl. Risk Chair: Elena Di Bernardino 16:20-16:35 Indranil Ghosh Some alternative bivariate Kumaraswamy-type distributions via copula with application in risk management 16:35-16:50 Sándor Guzmics A new exponential lifetime model and its resulting copula 16:50-17:05 Andreas Mändle Data driven partition-of-unity copulas with applications to risk management 17:05-17:20 Beatriz Vaz de Melo Spatial Copula Modelling of Crop Insurance Extreme Claims 17:20-17:35 Sabrina Mulinacci A shock model for systemically important institutions 17:35-17:50 Giorgia Rivieccio Copula quantile dependence for the analysis of multiple time series 8

9 Tuesday Time Speaker Title of the talk 09:00-10:00 Plenary Talk Chair: Manuel Úbeda Flores José J. Quesada Molina My meetings with Roger B. Nelsen 10:00-10:20 Carlo Sempi Ludger Rüschendorf Risk bounds with additional structural and dependence information 10:20-10:40 Coffee break 10:40-12:00 Copula models with Organizer & Chair: Johanna Nešlehová stochastic interpretation 10:40-11:00 Lei Hua Multivariate dependence modeling based on comonotonic factors 11:00-11:20 Johanna Nešlehová Reciprocal Archimedean Copulas 11:20-11:40 Ostap Okhrin A Nonparametric Change Point Model for Multivariate Statistical Process Control 11:40-12:00 Matthias Scherer Identifying tail-dependence and Bernoulli matrices in high-dimensions via linear programming 12:00-12:10 Mini-break 12:10-13:00 Plenary Talk Chair: Roger B. Nelsen Claudi Alsina The Charming Work of Roger B. Nelsen in Proofs without Words 13:00-14:20 Lunch break 14:20-15:40 Empirical copulas Organizer: Arturo Erdely. Chair: José-María González- Barrios 14:20-14:40 Emmanuel Ambriz Lobato Undirected graphical models via non-parametric conditional copulas 14:40-15:00 José-María González-Barrios Sample copula properties and a comparison to the empirical copula 15:00-15:20 Ricardo Hoyos-Argüelles Distributions of the random variables associated to the d- sample copula of order m under independence and weak convergence of the sample process 15:20-15:40 Coffee break 15:40-16:40 Vine Copulas Chair: Ostap Okhrin 15:40-15:55 Alex Donov Modelling dependence structure between stock indices using a second-order regime-switching vine copula 15:55-16:10 Ozan Evkaya Multivariate Analysis of Drought Characteristics using Vine Copula Approach 16:10-16:25 Diana Moreno Chavarro Vine copula model for spatial time series applied to daily mean temperature in Colombia 16:25-16:40 Marta Nai Ruscone Model based clustering for three-way data structures through a mixture of vine copulas 19:00-22:00 Social Event 9

10 Wednesday Time Speaker Title of the talk 09:00-10:00 Plenary Talk Chair: José J. Quesada Molina Carlo Sempi Quasi-copulas: A brief survey 10:00-10:20 Chair: Hans De Meyer Carles M. Cuadras Generating probability densities and copulas with distances and correlation functions 10:20-10:40 Coffee break 10:40-12:00 New trends in Copula Theory Organizer & Chair: Piotr Jaworski 10:40-11:00 Fabrizio Durante Spatially Homogeneous Copulas 11:00-11:20 Jean-David Fermanian On Kendall s regressions 11:20-11:40 Piotr Jaworski On the Conditional Value-at-Risk (CoVaR) in copula setting 11:40-12:00 Radko Mesiar On some recent copula constructions 12:00-12:15 Mini-break 12:15-13:00 Bayesian Chair: Roberto Ghiselli Ricci 12:15-12:30 Matthew Arvanitis A 4-Parameter Copula with Gamma Components 12:30-12:45 Jan Górecki Naive Bayes: Loosing Naivety with Copulas 12:45-13:00 Clara Grazian Approximate Bayesian inference in semiparametric copula models 13:00-14:20 Lunch break 14:20-15:35 Miscellaneous Chair: José María González Barrios 14:20-14:35 Yuri Salazar Flores The Existence of Tail dependence in the General Dependence Case and its Relationship with Domain of Attraction and Regular Variation Conditions for Copulas 14:35-14:50 María Isabel Ortego Watch out! Spurious copulas 14:50-15:05 Roberto Ghiselli Ricci Differential properties of copulas 15:05-15:20 Vadoud Najjari Copulas in stochastic frontier models by an application 15:20-15:35 Wilmer Pineda Ríos Galambos copula model for bivariate time series applied to daily mean temperature in Colombia: a bayesian approach 15:35-16:00 Coffee break 16:00-17:00 Theory and Appl. Chair: Fabrizio Durante 16:00-16:15 Martynas Manstavičius A class of bivariate copula mappings 16:15-16:30 Frank Oertel Sklar s Theorem and the Rüschendorf transform revisited - An analysis of right quantiles 16:30-16:45 Henrique Helfer Hoeltgebaum Theoretical results of the copula pertaining to the limit distribution of the two largest order statistics of iid samples 16:45-17:00 Silvia A. Osmetti PKL Optimal Design for Copula Models 17:00-17:10 Thank you very much! 10

11 s Special Talk Roger B. Nelsen, Lewis & Clark College (p. 12) Plenary Speakers Claudi Alsina, Universitat Politècnica de Catalunya José Juan Quesada Molina, Universidad de Granada (p. 13) Carlo Sempi, Università del Salento (p. 14) Special Sessions Empirical copulas (organizer: Arturo Erdely) Speakers: Emmanuel Ambriz (Mexico), José M. González-Barrios (Mexico), Ricardo Hoyos-Argüelles (Mexico), (p ). New trends in Copula Theory (organizer: Piotr Jaworski) Speakers: Fabrizio Durante (Italy), Jean-David Fermanian (France), Piotr Jaworski (Poland), Radko Mesiar (Slovakia), (p ). Recent copula constructions (organizer: Radko Mesiar) Speakers: José de Jesús Arias-García (Belgium), Tomáš Bacigál(Slovakia), Michal Dibala (Slovakia), L ubomíra Horanská (Slovakia), Matjaz Omladič (Slovenia), (p ). Copula models with stochastic interpretation (organizer: Johanna Neˇslehová) Speakers: Lei Hua (USA), Johanna Nešlehová (Canada), Ostap Okhrin (Germany), Matthias Scherer (Germany), (p ). Copulas with singular components (organizer: Wolfgang Trutschnig) Speakers: Elena di Bernardino (France), Giovanni Puccetti (Italy), Songkiat Sumetkijakan (Thailand), Wolfgang Trutschnig (Austria), (p ). Contributed Sessions Matthew Arvanitis (p. 35) Maximilian Coblenz (p. 36) Carles M. Cuadras (p. 37) Hans De Meyer (p. 38) Alex Donov (p. 39) Ozan Evkaya (p. 40) Sebastian Fuchs (p. 41) Roberto Ghiselli Ricci (p. 42) Indranil Ghosh (p. 43) Jan Górecki (p. 44) Clara Grazian (p. 45) Sándor Guzmics (p. 46) Henrique Helfer Hoeltgebaum (p. 47) Andreas Mändle (p. 48) Martynas Manstavičius (p. 49) Beatriz Vaz de Melo Mendes (p.50) Diana Moreno Chavarro (p. 51) Sabrina Mulinacci (p. 52) Vadoud Najjari (p. 53) Frank Oertel (p. 54) María Isabel Ortego (p. 55) Silvia Angela Osmetti (p. 56) Wilmer Pineda Ríos (p. 57) Mathias Pohl (p. 58) Giorgia Rivieccio (p. 59) Ludger Rüschendorf (p. 60) Marta Nai Ruscone (p. 61) Yuri Salazar Flores (p. 62) 11

12 Special Talk My Introduction to Copulas Roger B. Nelsen The author shares some memories of his early encounters with copulas and with individuals instrumental in the discovery of some of their fundamental properties Roger B. Nelsen, Lewis & Clark College, Portland, Oregon, USA address: 12

13 Plenary Speakers My meetings with Roger B. Nelsen José Juan Quesada Molina In this talk, we will briefly review our many meetings with Roger B. Nelsen, from August 1986 to the present day. A commented summary of the main results of Roger B. Nelsen and other colleagues work with the author is presented. Most of the mathematical results described here were obtained in the theory of copulas and quasi-copulas. José Juan Quesada Molina (corresponding author), Departamento de Matemática Aplicada, Universidad de Granada (Spain) address: jquesada@ugr.es 13

14 Quasi-copulas: A brief survey Carlo Sempi This talk is a survey of quasi-copulas. The genesis, the history and the properties of quasi-copulas are presented, with special emphasis on their characterisation and on recent results. The presentation will also highlight the key part played by Roger Nelsen in all stages of the development of the theory. Carlo Sempi (corresponding author), Dipartimento di Matematica e Fisica Ennio De Giorgi, Università del Salento, Lecce, Italy address: carlo.sempi@unisalento.it 14

15 Invited Sessions Undirected graphical models via non-parametric conditional copulas Emmanuel Ambriz Lobato Considering the restrictions in terms of the modelled dependence structures in undirected graphical models based on parametric copulas, we propose a nonparametric estimator for the undirected graph which takes advantage of the flexibility in the Bernstein copula models: the Bernstein estimator for the undirected graph. Using Joe s formula we develop an iterative algorithm for extracting observations from the conditional copulas involved in the undirected graph. This conditional observations are the basis for the inference of the graph from a hypothesis testing perspective. Acknowledging the multiplicity of tests for the estimation, we implemented a False Discovery Rate Control for dependent tests. The performance of the proposed estimator is evaluated by a simulation study. We conclude proposing future lines of research for the inference and applications of the undirected graphical models for high-dimensional data, i.e. when n << p. Emmanuel Ambriz Lobato, Centro de Investigación en Matemáticas (CIMAT), Jalisco s/n, Valenciana, 36240, Guanajuato, Gto., México. address: israel.ambriz@cimat.mx 15

16 Sample copula properties and a comparison to the empirical copula José M. González-Barrios Ricardo Hoyos-Argüelles The d-sample copula of order m, Cm n is a d-copula, which is an estimator of C (m) the checkerboard approximation of order m of a d-copula C, based on a sample of size n. As we will see in this talk, the estimator Cm n approaches C (m) for every m 2, when the sample size n increases. If m is relatively large, C (m) is a very good approximation of C. Hence, Cm n can be thought as a quasi-nonparametric method to estimate the real d-copula C, and it becomes a nonparametric estimator when we choose the order m. We will see that Cm, n the d-sample copula of order m, is simply the multilinear interpolation used in the proof of Sklar s Theorem, based on a pseudosample or modified sample. Then we will prove a Glivenko-Cantelli s Theorem for the sample d-copula Cm n and the real d-copula C, when m increases to infinity. We will also see that we can easily simulate from the sample d-copula Cm, n and that these simulated samples follow the same pattern of the original sample. We also provide evidence that it is possible to obtain a Glivenko-Cantelli s Theorem for the total variation distance between the checkerboard approximation C (m) and the sample copula Cm n for every m 2. We will also include examples with real data to study its behavior. Finally, we make a comparison of the supremum distance between the empirical copula C n and the real copula C, and the supremum distance between the real copula and the sample copula C n m, and we provide a method to estimate an adequate value of m associated to small sample sizes. José M. González-Barrios (corresponding author), Universidad Nacional Autónoma de México, IIMAS, Ciudad Universitaria, Circuito Escolar s/n, Mexico City, México address: gonzaba@sigma.iimas.unam.mx Ricardo Hoyos Argüelles, Universidad Nacional Autónoma de México, IIMAS, Ciudad Universitaria, Circuito Escolar s/n, Mexico City, México address: richoyos@hotmail.com 16

17 Distributions of the random variables associated to the d-sample copula of order m under independence and weak convergence of the sample process. Ricardo Hoyos-Argüelles José M. González-Barrios In the first part of this talk, following the ideas given by Deheuvels for the empirical copula, we first assume that we are sampling from the product copula, and we construct C n m the d-sample copula of order m. Then we study the distribution of the number of observations in each of the boxes generated by the regular partition of order m in [0, 1] 2, and then we give the associated moments and correlations. We also give generalizations of these results for the d-dimensional case, when d > 2. In the second part, we study the weak convergence of the sample process, that is, ( n Cm n C (m)), where C (m) is the checkerboard approximation of order m of the real copula C, when the partial derivatives of C are continuous, with or without the independence assumption. In this way we will have, for the sample copula, a weak convergence theorem similar to that for the empirical copula. In order to apply the results of weak convergence for the empirical process, we show that the sample copula can be represented as a linear functional of the empirical copula, and then we can apply the delta method. Finally, we perform several simulations of the sample process at a given point to find an adequate sample size that provides the convergence to a centered Gaussian process with a given variance-covariance structure. Ricardo Hoyos-Argüelles (corresponding author), Universidad Nacional Autónoma de México, IIMAS, Ciudad Universitaria, Circuito Escolar s/n, Mexico City, México address: richoyos@hotmail.com José M. González-Barrios, Universidad Nacional Autónoma de México, IIMAS, Ciudad Universitaria, Circuito Escolar s/n, Mexico City, México address: gonzaba@sigma.iimas.unam.mx 17

18 Spatially Homogeneous Copulas Fabrizio Durante Juan Fernández Sánchez Wolfgang Trutschnig In 1966, J.R. Brown introduced the following notion: A Markov operator on L 1 ([0, 1]) is called spatially homogeneous if it commutes with all rotations. Here, by exploiting the one-to-one correspondence between copulas and Markov operators, we consider spatially homogeneous copulas and present some of their properties. Specifically, we show how spatially homogeneous copulas are represented by probability measures on [0, 1). Moreover, we prove some symmetry properties and demonstrate how spatially homogeneous copulas can be used in order to construct copulas with surprisingly unusual properties. Finally, a generalization of spatially homogeneous copulas is studied and a characterization of this new family in terms of the Markov product of copulas is established. Fabrizio Durante (corresponding author), Dipartimento di Scienze dell Economia, Università del Salento, Lecce, Italy address: fabrizio.durante@unisalento.it J. Fernández Sánchez, Grupo de Investigación de Análisis Matemático, Universidad de Almería, La Cañada de San Urbano, Almería, Spain address: juanfernandez@ual.es W. Trutschnig, Department for Mathematics, University of Salzburg, Salzburg, Austria address: wolfgang@trutschnig.net 18

19 On Kendall s regressions Alexis Derumigny Jean-David Fermanian We review and introduce some semiparametric models that are based on conditional Kendall s tau. In the case of single-index models, we state some finite distance and asymptotic properties of the corresponding estimated parameters. Jean-David Fermanian (corresponding author), ENSAE-Crest, 3 avenue Pierre Larousse, Malakoff cedex, France address: Jean-David.Fermanian@ensae.fr Alexis Derumigny, Crest, 3 avenue Pierre Larousse, Malakoff cedex, France address: Alexis.DERUMIFNY@ensae.fr 19

20 On the Conditional Value-at-Risk (CoVaR) in copula setting Piotr Jaworski CoVaR (conditional Value at Risk) is a newly introduced risk measure which is oriented on systemic risk. If random variables X and Y are modelling our phenomena, say financial positions or gains from the investments, CoVaR of Y with respect to X is VaR of conditional Y. In more details CoV ar β (Y X) = V ar β (Y X E), where E, the Borel subset of the real line, is modelling some adverse event concerning X. Hence it is useful in stress testing, it measures how risky would be Y in case X is in trouble. Most often E consists of one point, an α quantile of X or a half-line on the left of it. For example the Modified Conditional-VaR at a level (α, β), which is a main objective of my presentation, is defined as VaR at level β of Y under the condition that X V ar α (X). When the distribution functions of X and Y are continuous the Modified CoVaR is equal to VaR of the (unconditional) Y at the level w, which depends only on the copula of the pair X, Y and levels α and β. In my presentation I will study to what extent the tail behaviour of the copula determines the limiting performance of the implied threshold w when the conditioning event is becoming more extreme. Basing on the differential characterization of copulas, this problem can be reduced to the study of the limiting properties of trajectories of the two-dimensional dynamical system in the presence of a saddle point. Piotr Jaworski (corresponding author), University of Warsaw, Poland address: P.Jaworski@mimuw.edu.pl 20

21 On some recent copula constructions Radko Mesiar Anna Kolesárová We recall and exemplify several recently introduced construction met-hods for bivariate copulas. In particular, we discuss: quadratic constructions of copulas; based on quadratic constructions, we introduce construction of gene-ralized Fairly Gumbel Morgenstern family; copulas derived from modular functions; constructions related to ultramodularity and Schur concavity; constructions based on n ary ultramodular quasi copulas. Note that the last introduced method is strongly related to the Archimax copulas, including EV copulas and conic copulas. Acknowledgements The support of the grant APVV is kindly announced. Radko Mesiar (corresponding author),faculty of Civil Engineering STU, Bratislava, Slovakia address: radko.mesiar@stuba.sk Anna Kolesárová, Faculty of Chemical and Food Technology STU, Bratislava, Slovakia address: anna.kolesarova@stuba.sk 21

22 Intermediate classes between quasi-copulas and copulas in higher dimensions J.J. Arias-García H. De Meyer R. Mesiar B. De Baets The notion of an n-quasi-copula was originally used to characterize operations on distribution functions that can or cannot be derived from operations on random variables, but was later characterized in terms of simpler analytical properties. Using the latter characterization, it has been shown that quasi-copulas play a vital role in the problem setting of finding the best possible bounds on a given subset of copulas of interest. In this talk, we recall some basic properties of n-quasi-copulas, their relationship with n-copulas and highlight some properties of 2-copulas that cannot be extended to higher dimensions. Then, we observe that as the dimensionality increases there are more intermediate classes between the class of n-quasi-copulas and the class of n-copulas. One such class consists of supermodular n-quasi-copulas, which coincides with the class of copulas for n = 2. We show that supermodular n-quasi-copulas have properties that are similar to those of 2-copulas that do not hold for higher-dimensional copulas. Finally, we use the newly introduced classes to generalize a volume-based characterization of bivariate quasi-copulas to higher dimensions. J.J. Arias-García (corresponding author), B. De Baets, KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Ghent, Belgium address:josedejesus.ariasgarcia@ugent.be, Bernard.DeBaets@ugent.be H. De Meyer, Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281 S9, B-9000 Ghent, Belgium address:hans.demeyer@ugent.be R. Mesiar (supported by grant APVV ), Department of Mathematics and Descriptive Geometry, Slovak University of Technology, Faculty of Civil Engineering, Radlinského 11, Bratislava, Slovak Republic address:radko.mesiar@stuba.sk 22

23 Some open problems in statistical interpretation of Williamson s n-transform Tomáš Bacigál We discuss several examples of connection between distributions of positively valued random variables (represented by distance function) and Archimedean copulas (represented by generator and it s inverse) through Williamson s transformation and Laplace transform, especially when arranged in a sequence. For instance, Williamson s n-transform (n = 2, 3,...) links the product copula with distribution of sum of n exponentially distributed independent random variables while the Laplace transformation links it to the most elementary distance function, the Dirac function. Naturally there appears a question: how can we use statistical properties of distance functions to draw statistical properties of copulas, and vice versa? Acknowledgment. The work of the first author was supported by the grant APVV Tomáš Bacigál (corresponding author), Slovak University of Technology in Bratislava, Radlinského 11 Bratislava, Slovakia address: tomas.bacigal@stuba.sk 23

24 Discrete copulas and maximal entropy principle Michal Dibala Mirko Navara The standard approach to fitting the parametric copula classes is widely used across various copula applications. The hidden assumption behind is the knowledge of dependence model for which we try to estimate the parameters. However, there are many applications where this approach is not desirable because of the lack of knowledge, insufficient data sample size, or a high dimension of the task. In cases where the whole given information is included in the data sample, the empirical or discrete copulas and their continous extensions can be used. Because of the incomplete information of copula values inbetween the known values, maximal entropy principle can also be applied. We propose an efficient algorithm for finding the maximal entropy copula fitted to given data. This appears to be also the maximally independent approximation. Besides, our algorithm does not lead to sparse distributions. The main advantage of this approach is the interpretability of the result. Acknowledgment. The work of the first author was supported by the grant APVV Michal Dibala (corresponding author), Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, , Slovakia address: dibala@math.sk Mirko Navara, Department of Cybernetics Faculty of Electrical Engineering, Czech Technical University, Technická 2, Praha 6, Czech Republic address: navara@cmp.felk.cvut.cz 24

25 On generalized symmetry of copulas L ubomíra Horanská Peter Sarkoci Construction of copulas based on (a, b)-transformations was deeply studied in (Horanská, A. Kolesárová, On (a, b)-transformations of conjunctive functions.fuzzy Sets and Systems (2017) doi: /j.fss ). Given a pair (a, b) [0, 1] 2, the (a, b)-transformation assigns to each copula C the copula C a,b defined by C a,b (x, y) = µ C ([a(1 x), x + a(1 x)] [b(1 y), y + b(1 y)]), where µ C denotes the probability measure induced by copula C. The (a, b)-transformations can be viewed as generalizations of the well-known survival and flipping constructions and can also be characterized by means of special measure-preserving transformations of the unit interval. Copulas invariant w.r.t. (a, b)-transformations are symmetric in some generalized sense. In our contribution we focus on finding invariant copulas using some tools of ergodic theory and iterated function systems. We also give a probabilistic interpretation of (a, b)- transformations. Peter Sarkoci gratefully aknowledges support of the Slovak Research and Development Agency under grant APVV and the project VEGA 2/0069/16. L ubomíra Horanská (corresponding author), Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Radlinského 9, Bratislava, Slovakia address: lubomira.horanska@stuba.sk Peter Sarkoci, Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, Bratislava, Slovakia address: peter.sarkoci@gmail.com 25

26 Inverse-maxmin copulas and their function in modelling quadrant subindependence Košir, Tomaž Omladič, Matjaž Dependence concepts play a crucial role in multivariate statistical literature since it was recognized that the independence assumption cannot describe conveniently the behavior of a stochastic system. The starting motivation of this paper is application to shock models, where (in the binary case) Marshall-Olkin and Marshall approach cover the survival of a two-component system subject to shocks. The case when components have a recovery option was covered by maxmin copulas that were introduced in a recent paper by Omladič, M., Ružić, N. (Shock models with recovery option via the maxmin copulas, Fuzzy Sets and Systems, 284 (2016), ) and further extended in a paper by Durante, F., Omladič, M., Oražem, L., and Ružić, N. (Shock models with dependence and asymmetric linkages to appear in Fuzzy Sets and Systems). Other applications of maxmin copulas were presented in the two cited papers. In this talk we propose a new approach to these copulas in which we reverse the meaning of one of the variables. The new class of copulas is then called the inverse-maxmin copulas. This methodology simplifies the technical details substantially, gives rise to a more intuitive view on the properties of these copulas and further extends the possibilities for applications. In particular the proposed strategy enables us to model a special concept of a stochastic relation which we call quadrant subindependence. This notion is a special case of negative quadrant dependence and should not be confused with the standard probability concept of subindependence. Omladič, Matjaž (corresponding author), Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia address: matjaz@omladic.net Košir, Tomaž, Department of Mathematics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia address: tomaz.kosir@fmf.uni-lj.si 26

27 Multivariate dependence modeling based on comonotonic factors Lei Hua Harry Joe Comonotonic latent variables are introduced into general factor models, in order to allow non-linear transformations of latent factors, so that various multivariate dependence structures can be captured. Through decomposing each univariate marginal into several components, and letting some components belong to different sets of comonotonic latent variables, a great variety of multivariate models can be constructed, and their induced copulas can be used to model various multivariate dependence structures. The paper focuses on an extension of Archimedean copulas constructed by Laplace Transforms of positive random variables. The corresponding comonotonic factor models with one set of comonotonic latent variables and multiple sets of comonotonic latent variables are studied, respectively. In particular, we propose several parametric comonotonic factor models that are useful in accommodating both within-group and between-group dependence with possible asymmetric tail dependence. Numerical methods for estimation with the resulting copula models are studied. There is an application using a dataset of body composition measurements to demonstrate the usefulness of the proposed parsimonious dependence models. Lei Hua (Speaker), Northern Illinois University, DeKalb, IL, United States address: Lhua@niu.edu Harry Joe, University of British Columbia, Vancouver, BC, Canada 27

28 Reciprocal Archimedean Copulas Johanna G. Nešlehová Members of the well-known family of bivariate Galambos copulas can be expressed in a closed form in terms of the univariate Fréchet distribution. This formula extends to any dimension and can be used to define a whole new class of tractable multivariate copulas that are generated by suitable univariate distributions. In this presentation, I will derive the necessary and sufficient conditions on the underlying univariate distribution that ensure that the resulting copula indeed exists. I will also show that these new copulas are in fact dependence structures of certain max-id distributions with l 1 -norm symmetric exponent measure. Basic dependence properties of this new class will be investigated along with an efficient algorithm for random number generation. This is joint work with Christian Genest and Louis-Paul Rivest. Johanna G. Nešlehová, McGill University, 805, rue Sherbrooke Ouest, Montréal (QC), Canada H3A 0B9 address: johanna.neslehova@mcgill.ca 28

29 A Nonparametric Change Point Model for Multivariate Statistical Process Control Ostap Okhrin Ya-Fei Xu This article presents a nonparametric change point model for multivariate statistical process control (MSPC) based on maximum energy test. Three highlights are included. Firstly, it is a nonparametric change point model which requires no preknowledge on the process compared to the classical parametric control chart. Secondly, it is oriented to Phase II change point detection which is central for real time surveillance of stream data and can be applied extensively, e.g. in industrial quality control, finance, medical science, geology et. al. Thirdly the model is designed for multivariate time series, which is more practical and informative for catching the essence of data as a whole than univariate time series. In simulation study the metrics for in-control and out-of-control average run length (IC-ARL, OC-ARL) of the model have been investigated in context of mean shift, variance shift, tail shift and distribution shift, especially the tail shift and the distribution shift are studied by R-Vine copulas. In empirical study we employed a multivariate time series data set from U.S. ETF market to analyze the model s performance for financial surveillance especially in financial turmoil periods. The results from simulation and empirical study showed that the model has great potential in sequential detection of multiple change points for high dimensional time series. In this article authors contributed an R package EnergyOnlineCPM. Ostap Okhrin (corresponding author), Chair of Econometrics and Statistics, Dresden University of Technology, Germany Ya-Fei Xu, School of Business and Economics, Humboldt-Universität zu Berlin, Germany 29

30 Identifying tail-dependence and Bernoulli matrices in high-dimensions via linear programming Daniel Krause Matthias Scherer Jonas Schwinn Ralf Werner In the context of market-, credit-, and operational risk, stochastic models allowing for tail dependence are considered indispensable in modern risk-management. Being difficult to estimate, it is often a matter of expert judgment to define a matrix of pairwise tail-dependence coefficients. Given such a matrix, however, it is rather difficult to decide if (i) the given matrix is indeed a possible tail-dependence matrix, and (ii) how a stochastic model can be constructed representing it. These problems, and the one-to-one connection to Bernoulli matrices, has already been studied on a theoretical level, but efficient numerical tests are rare. We add to the existing literature by exploiting the polyhedral geometry of the set of Bernoulli matrices. This allows to translate the above questions into a linear optimization problem with exponentially many variables. Moreover, we partially overcome the curse of dimensionality by a specific column generation ansatz. Daniel Krause and Matthias Scherer (corresponding author), Lehrstuhl für Finanzmathematik, Technische Universität München, Germany address: scherer@tum.de Jonas Schwinn and Ralf Werner, Institut für Mathematik, Universität Augsburg, Germany 30

31 Copulas and (some) multivariate risk models Elena Di Bernardino In the last decade, much research has been devoted to construction of risk measures that account both for marginal effects and dependence between risks. This work belongs to this large body of literature and it can be seen as a further effort in order to extend risk analysis in a multidimensional setting by using copula theory. We try to reach a contribution in the following directions: (1) showing the importance and explaining the usefulness of the multivariate copula framework in risk analysis, (2) fitting the appropriate multivariate distribution in order to model risks and (3) defining multivariate risk measures. For instance we propose distortion/transformation copula models which allow changing the copula in multivariate tails or in the center of multivariate distributions. These modifications can distort an initial copula in order to leave for instance the class of Archimedean copulas. All these possibilities give suitable flexibility in order to model multivariate dependence, which is essential in the construction of risk models. Elena Di Bernardino (corresponding author), Conservatoire national des arts et mãľtiers (CNAM), 292 rue Saint-Martin, Paris Cedex 03, France address: elena.di bernardino@cnam.fr 31

32 Multivariate notions of extremal dependence and the swapping algorithm Giovanni Puccetti First, we consider several multivariate extensions of comonotonicity and countermonotonicity. We show that naive extensions do not enjoy some of the main properties of the univariate concepts. In order to have these properties, more structures are needed than in the univariate case and we define extremal multivariate dependence concepts based on optimization properties. Optimal measures maximizing (minimizing) the expected inner product of two marginals are called c-co(unter)monotonic, as they generalize the case of maximal (minimal) correlation to multivariate marginal distributions. Then, we introduce a new algorithm, called the swapping algorithm, to approximate numerically the minimal and maximal expected inner product of two random vectors with given marginal distributions. As a direct application to mass transportation problems, the algorithm computes an approximation of the L2- Wasserstein distance between two multivariate measures. Department of Economics, Management and Quantitative Methods, University of Milan Via Conservatorio 7, Milano, Italy address: giovanni.puccetti@unimi.it 32

33 Non-atomic copulas Songkiat Sumetkijakan Tanes Wongyang A partial factorization of every non-atomic copula/markov operator suggests its close ties with implicit dependence copulas. Based on their associated σ-algebras, we investigate their relationships further via an equivalence relation on the non-atomic copulas. Implicit dependence copulas are shown to be natural representatives of the equivalence classes. A partial ordering of these classes should lead to a natural condition on measures of dependence. Songkiat Sumetkijakan (corresponding author), Chulalongkorn University, Phayathai Road, Patumwan, Bangkok, Thailand address: someone1@mail.com Tanes Wongyang, Chulalongkorn University, Phayathai Road, Patumwan, Bangkok, Thailand address: t.wongyang@gmail.com 33

34 Complete dependence everywhere? Wolfgang Trutschnig Describing the situation of full predictability of a random variable Y given the value of another random variable X, the notion of complete dependence might seem far too restrictive to be of any practical importance at first glance. Recent results have shown, however, that complete dependence naturally appears in various settings. My talk will focus on dependence measures strictly separating extreme dependence concepts, on a problem related to joint-default maximization, on a question from uniform distribution theory, and on the relationship between the two most well-known measures of concordance, Kendall s τ and Spearman s ρ. A short excursion to topology showing that complete dependence is not atypical at all concludes the talk. Wolfgang Trutschnig, Department for Mathematics, University Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria address: wolfgang@trutschnig.net 34

35 Contributed Sessions A 4-Parameter Copula with Gamma Components Matthew Arvanitis Muli-parameter copulas are seldom applied in practice. In this work, a flexible family of 4-parameter copulas first introduced by Arnold & Ng will be discussed in detail. Also presented will be a likelihood-free Bayesian method of parameter estimation which applies Sklar s Theorem to obtain a prior distribution with an interpretable dependence structure, while also preserving non-informative marginals. In addition, some model-building strategies will be discussed relating to applications of this family s numerous sub-models. Matthew Arvanitis (corresponding author), University of California, Riverside, 900 University Ave., Riverside, CA 92521, USA address: matthew.arvanitis@ .ucr.edu 35

36 The Length of Copula Level Curves Maximilian Coblenz Oliver Grothe We investigate level curves of bivariate copulas more closely. In particular, we consider the length of these curves and formulate weak conditions on the copulas so that the length is finite and exists. We define the length profile L C (t) that maps level t [0, 1] to the length of its level curve for copula C. We show continuity and differentiability of L C (t) and provide pointwise boundaries. We investigate further properties of the length profile for different copula families and provide formulas for some Archimedean families. L C (t) is informative with respect to the dependence structure of C and we relate it to different measures of concordance and the tail dependence coefficients. Moreover, a graphical tool based on the length profile is outlined, which can help researchers when exploring the dependence structure of data. Finally, we extend our results to dimensions higher than two. Maximilian Coblenz (corresponding author), Karlsruhe Institute of Technology, Institute of Operations Research, Analytics and Statistics, Kaiserstr. 12, Karlsruhe, Germany address: maximilian.coblenz@kit.edu Oliver Grothe, Karlsruhe Institute of Technology, Institute of Operations Research, Analytics and Statistics, Kaiserstr. 12, Karlsruhe, Germany address: oliver.grothe@kit.edu 36

37 Generating probability densities and copulas with distances and correlation functions Cuadras, C. M. Ortego, M. I. We show some properties of the geometric variability and the proximity function of a set in relation with a distance. The Shannon entropy is the lower bound for the geometric variability. Then we propose a procedure for constructing probability densities with distances and symmetric functions. Also using distances, we define the geometric dimensionality of a copula and consider the case of continuous dimensionality rather than countable. Then Lancaster expansion of a copula in terms of series is generalized and expressed in terms of integrals. As a consequence, we define a class of correlation functions and propose a method for generating a wide family of copulas. Cuadras, C. M. (corresponding author), Universitat de Barcelona, Barcelona (Spain) address: ccuadras@ub.edu Ortego, M. I., Universitat Politècnica de Catalunya-BarcelonaTECH, Jordi Girona 1, Barcelona (Spain) address: ma.isabel.ortego@upc.edu 37

38 Symmetry properties of a (quasi-)copula with respect to a curve H. De Meyer B. De Baets We extend the seminal work of Roger Nelsen on symmetry-related properties and the degree of asymmetry of copulas, by reattributing the role the diagonal plays as axis of symmetry to a continuous strictly increasing curve in the unit square. Firstly, we provide a probabilistic interpretation for the curvilinear section of a copula, and we highlight the geometrical notion and probabilistic interpretation of the degree of symmetry of a (quasi-)copula with respect to a curve. Next, we provide a measure for quantifying to what extent a (quasi-)copula can be regarded asymmetric with respect to a curve. The main part of the contribution is devoted to the derivation of a sharp lower and sharp upper bound on the degree of asymmetry a (quasi-)copula can possess with respect to a curve. It is shown that the lower bound is attained by any symmetric (quasi-)copula (i.e. symmetric w.r.t. the diagonal of the unit square). The upper bound is shown to be equal to the maximum of the fixed points of two different functions depending on the curve. We also lay bare some interesting properties of (quasi-)copulas that have the maximal degree of asymmetry w.r.t. a curve. We conclude with pointing to open questions and possible patterns for future research. H. De Meyer(corresponding author), Department of Applied Mathematics, Computer Science and Statistics, GHent University, Krijgslaan 281 S9, B Gent, Belgium address: hans.demeyer@ugent.be B. De Baets, KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B Ghent, Belgium address: bernard.debaets@ugent.be 38

39 Modelling dependence structure between stock indices using a second-order regime-switching vine copula Alex Donov In this study first-order regime-switching vine copula is extended to a second-order regime-switching vine copula. This model is applied to returns of the S&P500, FTSE100 and DAX stock indices. Empirically, two regimes are identified: one of high and one of low dependence. In order to allow asymmetries in the upper and lower tail-dependence, we have compared 10 vine copulas with different tail behaviour. The dependence parameters are held constant within each regime. The standard errors of the estimates are computed using Godambe information matrix. For the sample period under investigation, based on the principle of maximum likelihood, Gaussian bivariate copulas were selected as main building blocks in the vine copula specification. The model is then compared against the benchmark model in which regime variable follows first-order process. The information criteria such as AIC and BIC suggest that the second-order regime-switching vine copula may be a better choice. Alex Donov, University of Essex,Wivenhoe Park, Colchester, CO4 3SQ, UK address: adonae@essex.ac.uk 39

40 Multivariate Analysis of Drought Characteristics using Vine Copula Approach Ozan Evkaya Ceylan Yozgatlıgil A. Sevtap Selcuk-Kestel Drought, occurs over different time scales, is one of the most devastating natural hazards having economic, environmental and social impacts. Drought characteristics provide useful information for decision makers to manage drought related risks. Especially, drought duration, severity and peak intensity are important variables in various research fields. This paper mainly investigate the multivariate probabilistic analysis of different drought characteristics. For this research, widely used drought indices, namely SPI, SPEI and RDI were considered over distinct time scales. For the construction of multivariate distribution of drought duration, severity and peak intensity, vine copula framework was utilized based on a wide range of copula families. Vine copula model was evaluated to identify the dependence structure in each drought state over the data set of 1950 and 2010 for Kulu, Turkey, as a local station. Additionally, drought probabilities and return periods were calculated based on the derived vine copula model. Ozan Evkaya (corresponding author), Atilim University, Kızılcasar Mah. Incek Ankara, Turkey ozanevkaya@gmail.com Ceylan Yozgatlıgil, Middle East Technical University, Üniversiteler Mahallesi, Dumlupınar Bulvarı No:1 Ankara, Turkey ceylan@metu.edu.tr A. Sevtap Selcuk-Kestel, Middle East Technical University, Üniversiteler Mahallesi, Dumlupınar Bulvarı No:1 Ankara, Turkey skestel@metu.edu.tr 40

41 Estimators for a Class of Bivariate Measures of Concordance for Copulas Sebastian Fuchs Klaus D. Schmidt We propose and study estimators for a wide class of bivariate measures of concordance for copulas. These measures of concordance are generated by a copula and generalize Spearman s rho and Gini s gamma. In the case of Spearman s rho and Gini s gamma the estimators turn out to be the usual sample versions of these measures of concordance. Sebastian Fuchs (corresponding author), Technische Universität Dresden, Dresden, Germany address: sebastian.fuchs1@tu-dresden.de Klaus D. Schmidt, Technische Universität Dresden, Dresden, Germany address: klaus.d.schmidt@tu-dresden.de 41

42 Differential properties of copulas Roberto Ghiselli Ricci We focus on the property of 2-increasingness, showing a general characterization result for any real mapping defined on some subdomain of the real plane. Then, we illustrate how to use it for characterizing bivariate copulas. In particular, we rigorously show the existence, usually taken for granted or not well explained, of mappings obtained as particular restrictions of the first partial derivatives of a quasicopula. Our approach relies upon a nice application of the celebrated Rademacher s theorem. Actually, we will show that it is possible to achieve the same theoretical result even weakening the assumptions required for a candidate to be a bivariate copula: in this case, we have to replace Rademacher s theorem with a (little-known) remarkable result of real analysis due to James Serrin. Eventually, we will extend our arguments in order to cover copulas of higher dimension. Roberto Ghiselli Ricci, University of Ferrara, Via Voltapaletto 11, Ferrara, Italy address: ghsrrt@unife.it 42