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Measuring properties in pixels to reveal an image description is of great essential for image processing. One of obstacles in image splicing detection is an unavailability of complete-reference image. In this study, a statistical method is proposed using a cumulative joint distribution of colours pixel. Consider an image consisting of particular pixels with RGB (Red, Green, Blue) components, each colour provides unique behaviour of probability function that implicitly sticking in marginal distributions. Copulas provide a cumulative joint distribution that consist of the univariate marginals and load their dependency structure, since the deficiency of linear dependency under non-linear strictly transformation. Copulas can determine the distinguish the cumulative joint distribution in the edge boundaries of image splicing. According to experimental results, Gaussian and Archimedean copulas provide deeper description to detect the contrast copula parameters of splicing image distribution.

The design of timber connections with dowel-type fasteners is dependent on the knowledge of their mechanical behavior and failure modes. Concerning the main design parameters, the timber embedment strength and the dowel bending moment capacity are the parameters that govern the load-carrying capacity. The correlation between the timber embedment strength and the dowel bending moment capacity has not been sufficiently addressed in the literature yet. However, since they both share a common dependency to the timber density, they are probably correlated. To investigate this, traditional distribution fitting procedures, as well as copula functions, are implemented to consider the correlation between them. By doing so, it is aimed to evaluate the effectiveness of the different approaches in describing the dependence structure of the variables and their influence on the structural reliability. It was found that, for single dowel-type connections, the impact of the copulas on the results is small. It is indicated that, unless significantly nonlinear correlations exist among the data, the results obtained by applying different copula functions will probably be very close., FCT - Fundação para a Ciência e a Tecnologia(POCI-01– 0145-FEDER-032031). FEDER funds through the Competitivity and Internationalization Operational Programme COMPETE, Portugal 2020, and by national funds through FCT – Foundation for Science and Technology within the scope of the Timquake project POCI-01–0145-FEDER-032031. The authors also thank the Brazilian Research Council (CNPq) for sponsoring this research through the grant 302489/2017–7 and through the scholarship grant conceded to the first author

Up to this point, we have mainly focused our efforts on univariate distributions. This was mostly just to keep the notation simple. Multivariate data, however, appear often in practice and multivariate distributions are eminently useful and important. It is time now to formalise multivariate distributions and explicitly discuss models for multivariate observations. The workhorse in this field is certainly the multivariate normal distribution, which we will explore in depth in Sect. 10.1. Beyond the normal distribution, copula-based distributions have been a hot topic in recent years. Copulas allow for complex dependence structures and this chapter provides a short introduction to the basic ideas of copulas in Sect. 10.2. Besides modelling multivariate data, it often occurs that extreme events are of interest, e.g. a maximal loss or a minimal supply level. This is addressed in Sect. 10.3, where we introduce extreme value distributions.

The framework of this chapter is to introduce a new efficient Blind Source Separation (BSS) method that handles mixtures of noise contaminated independent/dependent sources. This approach is based on the minimization of a regularized criterion. Specifically, it consists in combining the total variation method to de-noise the observations, with the Kullbak-Leibler divergence between the copula densities to separate the mixtures. This algorithm has shown its effectiveness and efficiency toward the noisy dependent/independent sources and also surpassed the standard BSS algorithms through different experimental results.

In this chapter, we will be introducing a new efficient Blind Source Separation (BSS) method that handles mixtures of independent and dependent sources. In order to achieve that, we minimize a criterion based on Kullback-Lebiler divergence to set apart the observed mixtures of either dependent or independent sources using copulas as a tool to model this dependency. The proposed algorithm utilizes a gradient descent method to retrieve the signal sources. The efficiency and robustness of this approach is showcased through different experimental results.

Copulas are used to create joint distributions with specified marginal distributions. The copula models the dependence between the corresponding marginal random variables. In the normal case, the multivariate normal distribution is a natural choice of joint distribution with normal marginals and its covariance matrix parameterizes the dependence between the individual marginal normals. But how would we specify a joint distribution for a normal and a half-normal, where these two random variables are allowed to be dependent? We can do this using copulas.

The structure of dependence between the level of economic development and investment provision of Russian regions in 2012–2016 is studied. Investment provision and the level of economic development in the regions are described by fixed capital investment (FCI) per capita and gross regional product (GRP) per capita, respectively. The copula method is used to model the dependency structure. It is established that the probabilistic structure of the relationship of the studied indicators can be described by a Gumbel copula in each year of the period 2012–2016, with the exception of 2014. The Gumbel copula in 2012–2013 transformed into Frank copula in 2014. There are no significant changes in the Gumbel copulas parameters in 2012–2013 and 2015–2016. It is concluded that the probabilistic structure describing by the Gumbel copula remains stable during the observed period. It is established that the Gumbel copula describes a sufficiently strong relationship between the studied parameters and has the asymptotic dependence in the upper tail of the distribution.

This paper investigates the dependence structure among nominal crude oil (WTI), gold, and specific U.S. dollar against four major currencies (Euro, British Pound, Japanese Yen and Canadian Dollar) on a daily basis over the last decade. In order to capture the tail dependence between commodity market and USD exchange rates, we apply both bivariate zero tail and tail copulas, combined with the AR-GARCH marginal distribution for gold, oil and exchange rates daily returns. The primary findings are as follows. Firstly, based on the concordance and correlation coefficient, we find that there is a positive correlation between gold and crude oil prices, and a negative dependence between gold and currencies as well as oil and currencies. Secondly, the crude oil price can be viewed as a short term indicator in the exchange rates movement; the crude oil price also can be viewed as a short term descend indicator of gold price, while the gold price is an short term rise indicator of oil price. Thirdly, small degree of conditional extreme tail dependence for all considered pairs are observed. Our results provide useful information in portfolio diversification, asset allocation and risk management for investors and researchers.

The paper considers the problem of establishing data support for the simplifying assumption (SA) in a bivariate conditional copula model. It is known that SA greatly simplifies the inference for a conditional copula model, but standard tools and methods for testing SA in a Bayesian setting tend to not provide reliable results. After splitting the observed data into training and test sets, the method proposed will use a flexible Bayesian model fit to the training data to define tests based on randomization and standard asymptotic theory. Its performance is studied using simulated data. The paper’s supplementary material also discusses theoretical justification for the method and implementations in alternative models of interest, e.g. Gaussian, Logistic and Quantile regressions.

Definitions and basic properties of bivariate Archimedean copula models for survival data are reviewed with an emphasis on their motivation via frailties. I present some new characterization results for Archimedean copula models based on a notion I call semi-invariance under truncation.

Air quality is a common cause for respiratory health problems. It shows high temporal and spatial variability within urban areas and currently sensors are installed to monitor air quality. The objective of this paper is to investigate its spatial variability during peak hours. Spatial statistical methods are based upon copula theory, integrating distributions from different pollutants and at different locations. In this paper attention focused on PM1 as one of the neglected components of air quality so far. Using observations from the Netherlands, we compared two hours: the first hour of the New Year, and an hour with a high traffic congestion. We investigated the size of the sensor network by analyzing observations from a city with 35 sensors with a city with four sensors and a city with one sensor. In the absence of an environmental standard for PM1, the paper defined a threshold related to existing thresholds of PM2.5 and PM10. Results showed the adequacy of the large network, generating a varying pattern during the high peak hour, whereas in cities with less sensors both the spatial spread and possibly large values are missed. In particular the first hour of the New Year showed large concentrations and high probabilities that a threshold was exceeded, whereas the second peak hour showed values well below the threshold. We conclude that the mapping of PM1 concentrations can well be done by copula interpolation. Using one or three sensors may save expenses, but this is at the cost of missing extreme values as well as spatial patterns. Both are potentially important for public health measures.

In the paper Jin et al. [8] the authors introduced a generalized phi-transformation of aggregation functions. This is a kind of two-step aggregation. This transformation was further developed in Jin et al. [9] into a Generalized-Convex-Sum-Transformation. A special case of the proposed Generalized-Convex-Sum-Transformation is the well-known *-product, also known as the Darsow product of copulas. This approach covers also the discrete Choquet integral. In this paper we study the monotone systems of functions, particularly the case when functions in these systems are just two-valued.

One of important classical reliability problems can be formulated as follows. Given a k-component system (k = 2, 3, …) with series reliability structure where the life-times X1, …, Xk of the components are nonnegative stochastically dependent random variables. In order to determine the system’s (as a whole) reliability function as well as for some system maintenance analysis, one needs to find or construct a proper joint probability distribution of the random vector (X1, …,Xk) which may be expressed in terms of the joint reliability (survival) function. Numerous particular solutions for this problem are present in the literature, to mention only [Freund in J Am Stat Assoc 56:971–77, 1961 1, Gumbel in J Am Stat Assoc 55:698–707, 1960 2, Marshall and Olkin in J Appl Probab 4:291–303, 1967 3]. Many other, not directly associated with reliability, k-variate probability distributions were invented [Kotz et al. in Continuous Multivariate Distributions, Wiley, New York, 2000 4]. Some of them later turned out to be applicable to the above considered reliability problem. However, the need for proper models still highly exceeds the existing supply. In this chapter we present not only particular bivariate and k-variate new models, but, first of all, a general method for their construction competitive to the copula methodology [Sklar A in Fonctions de repartition a n dimensions et leurs marges, Publications de l’Institut de Statistique de l’Universite de Paris, pp. 229–231, 1959 5]. The method follows the invented universal representation of any bivariate and k-variate survival function different from the corresponding copula representation. A comparison of our representation with the one given by copulas is provided. Also, some new bivariate models for 2-component series systems are presented. Possible applications of our models and methods may go far beyond the reliability context, especially toward bio-medical and econometric areas.

In this paper we focus our attention on multi-dimensional copula models of returns of the indexes of selected prominent international financial markets. Our modeling results, based on elliptic copulas, 7-dimensional vine copulas and hierarchical Archimedean copulas demonstrate a dominant role of the SPX index among the considered major stock indexes (mainly at the first tree of the optimal vine copulas). Some interesting weaker conditional dependencies can be detected at it’s highest trees. Interestingly, while global optimal model (for the whole period of 277 months) belong to the Student class, the optimal local models can be found (with very minor differences in the values of GoF test statistic) in the classes of vine and hierarchical Archimedean copulas. The dominance of these models is most striking over the interval of the financial market crisis, where the quality of the best Student class model was providing a substantially poorer fit.

In the paper, we introduce a new class of star-shaped distributions where a special copula model is used for the distribution of the spherical angles. Extended Fairlie–Gumbel–Morgenstern families of copulas have a lot of advantages in this context and are deployed here.

Part of the Lecture Notes in Computer Science book series (LNCS, volume 11767)., We propose and demonstrate a joint model of anatomical shapes, image features and clinical indicators for statistical shape modeling and medical image analysis. The key idea is to employ a copula model to separate the joint dependency structure from the marginal distributions of variables of interest. This separation provides flexibility on the assumptions made during the modeling process. The proposed method can handle binary, discrete, ordinal and continuous variables. We demonstrate a simple and efficient way to include binary, discrete and ordinal variables into the modeling. We build Bayesian conditional models based on observed partial clinical indicators, features or shape based on Gaussian processes capturing the dependency structure. We apply the proposed method on a stroke dataset to jointly model the shape of the lateral ventricles, the spatial distribution of the white matter hyperintensity associated with periventricular white matter disease, and clinical indicators. The proposed method yields interpretable joint models for data exploration and patient-specific statistical shape models for medical image analysis., NIH/NIBIB/NAC (Grant P41EB015902), NIH/NINDS (Grant R01NS086905)

There are three major construction approaches of copulas. One arising from applying the probability integral transform (see Definition 1.3) to each margin of known multivariate distributions and one to use generator functions. The first approach applied to elliptical distributions yields the class of elliptical copulas. With the second approach, we obtain the class of Archimedean copulas. The well-known examples of this class are the Clayton, Gumbel, Frank, and Joe copula families. The third approach arises from extensions of univariate extreme-value theory to higher dimensions.

The dependence measures, like for example Kendall tau, Spearman rho, Blomquist beta or tail dependence coefficient, are the main numerical characterization of Bivariate Extreme Value Copulas. Such copulas are characterized by a function on the unit segment, called a Pickands dependence function, which is convex and comprised between two bounds. We identify the smallest possible compact sets containing the graphs of all Pickands dependence functions whose corresponding bivariate extreme-value copula has fixed values of given dependence measures. Moreover we provide the bounds for such bivariate extreme-value copulas.

This chapter reveals the crucial role that copulas play in MEVT. The D-norm approach again proves to be quite a helpful tool. In particular, it turns out that a multivariate df F is in the domain of attraction of a multivariate EVD iff this is true for the univariate margins of F together with the condition that the copula of F in its upper tail is close to that of a generalized Pareto copula. As a consequence, MEVT actually means extreme value theory for copulas.

In the last chapter, we have seen how to design very flexible multivariate copulas. To make them useful for practise, we have now to tackle the problem of parameter estimation. For this, we will assume that the simplifying assumption holds throughout. We also suppose, that the vine copula model is completely specified by a given vine tree sequence, as well as, the pair copula families associated to each edge in the vine tree sequence. The problem of how to choose the pair copula families and how to select the vine structure will be treated in subsequent chapters.

We analyzed quarterly seasonally-filtered time series (OECD link) of GDP in EURO per capita for the V4 countries (Visegrad treaty - Czech republic, Hungary, Poland, Slovakia) and Germany in the period 1996/Q1 – 2018/Q1. First, ARIMA models were used to clear the temporal dependence, then marginal distribution functions were utilized to standardize the data such that it is U(0,1) distributed. Dependence among the 5 random variables was analyzed in terms of correlation strength and joint distribution modeled by elliptical, vine and factor copulas, both in the whole 22 years period and 6-year rolling windows with 3-year overlaps. The choice of such different model classes allows us study the nature of underlying dependence structure from different standpoints.

Copula is a distribution function on the unit hypercube with uniform margins. The margin is directly related to the stochastic behaviour of one variable, while joint distribution function covers the holistic character of more. In multivariate (and particularly bivariate) analysis, using copulas is an elegant way to solve the missing information problem between joint distribution function and the total of the margins. Hereby, the intention of this paper is twofold. Firstly, the paper intends to emphasize the advantages of copulas in practice. In order to encourage potential researchers to diversify their subject of work with these functions, authors give the essential introductory details for a clear understanding of copulas associated with their basic mathematical and statistical preliminaries. Secondly, the study exemplifies the practical usage of copulas in statistical process control area. In this context, process parameters are estimated in order to calculate the control limits of a typical Shewhart type control chart. Parameter estimation is performed by Maximum Likelihood Estimation (MLE) for the bivariate Clayton copula in univariate AR (1) time series with several different levels of high dependence. Since monitoring autocorrelated data in control charts is known as being one of the main causes of producing tighter control limits than required, false alarm rate may be increased and accordingly, the performance of control charts may be dramatically decreased. This study shows that copulas may alternatively be used for getting the same or little wider acceptable region between upper and lower limits. This recognition of the properness of copulas may help to decrease some of the negative effects of dependent data being monitored on charts for further studies.

In this paper, a flexible method for constructing bivariate copulas is provided, which is a generalization of the so-called “gluing method” and “rectangular patchwork” constructions. A probabilistic interpretation of the construction is provided through the concept of the threshold copula. Properties of the construction and best-possible bounds of copulas with given threshold copulas are investigated. Examples are given for the illustration of our results.

Measuring dependence dynamics between oil and stock markets has attracted special attention from researchers in financial econometrics literature recently. This study aims to investigate the time-varying dependence between individual G7 stock market and Brent crude oil market. The Markov-switching time-varying copula is employed to examine the structural change in the time-varying dependence as well as explain the relationship between them. Our findings show evidence of regime shifts in the dependence structure between stock and oil returns only in United States and France. Moreover, both the Clayton and Gumbel highly outperform the symmetric elliptical copulas, which suggests the asymmetry between oil and stock returns in G7 countries. We also find the positive dependence between oil and stock indices for most cases.

This paper aims to use the Bayesian estimation as an alternative method for formulating and estimating mixed copula models. This method has claimed to be more efficient than the conventional maximum likelihood estimator as it can deal with the high dimension copula and large parameter estimates under limited sample. In this study, we present various mixed copula functions constructed from both Elliptical and Archimedean copulas. We employ a simulation study to investigate the performance of this estimator for comparison with the maximum likelihood estimator. The results show that the Bayesian estimation is considerably more accurate than maximum likelihood estimator in various scenarios. Finally, we extend the Bayesian mixed copula to the real data and show that our approach perform well in this real data analysis.

The purpose of this paper is to investigate the effects from natural rubber supply in seven ASEAN countries (i.e. Cambodia, Malaysia, Myanmar, Philippines, Singapore, Thailand and Vietnam) over the period of 2006–2017 collected by CEIC data. Methodologically, data classifications were processed by entropy analyses, and the structural dependent estimation is provided by copula models. The empirical results of econometric analyses represent countries which are a market leader and market followers. Additionally, the results would be useful for policies development regarding to the natural rubber productions, developments, and promotions.

This paper applies the concept of copula based GARCH to the problem of finding the efficient frontier associated with the risk-return portfolio optimization model. Various families of the mixed copula structures, both Elliptical and Archimedean copulas, are proposed to deal with the complicated dependence structure in the financial portfolio. We then apply these mixed copula models to investigate the portfolio risk composed of stock, oil and gold returns and also compare the mixed copula with the conventional copula in order to show the better performance of the mixed copula in terms of the lowest AIC and BIC. The empirical results show that, compared with traditional methods, the mixed Gaussian and Student-t copula model describes the dependence structure of the portfolio return series more successfully. Finally, we apply the best fitted model to do the Monte Carlo simulation for constructing the efficient frontier and find the optimal investment using the minimized variance and tangency portfolio approaches.

Copulas are useful tools for formalizing the dependence structure between variables. They have proven to be very valuable in economics, where the dependence plays a key role. In this chapter, we use copulas to analyze the dependence between inflation and US/Euro exchange rates in the Euro area, during different periods. We first explore the dependence between the variables using a nonparametric approach. Then, we select an appropriate parametric copula for each period. Results confirm the sensibility of copulas to macroeconomic fluctuations that occur during the analyzed periods.

The increase in drought features in North Africa and Mediterranean regions is one of the most threatening natural hazards related to climate change. The spatial extent of this phenomenon in the North-West of Algeria has caused serious economic, social, and environmental impact with enormous losses. In this study, the return periods of drought events characterized by high severities and the three main droughts characteristics (duration, severity, and intensity) in the future under two Representative Concentration Pathway (RCP4.5 and RCP8.5) over two representative watersheds in North-West Algeria have been estimated In order to plan and manage the water resources under drought conditions.

An importance sampling approach for sampling from copula models is introduced. The proposed algorithm improves Monte Carlo estimators when the functional of interest depends mainly on the behaviour of the underlying random vector when at least one of its components is large. Such problems often arise from dependence models in finance and insurance. The importance sampling framework we propose is particularly easy to implement for Archimedean copulas. We also show how the proposal distribution of our algorithm can be optimized by making a connection with stratified sampling. In a case study inspired by a typical insurance application, we obtain variance reduction factors sometimes larger than 1000 in comparison to standard Monte Carlo estimators when both importance sampling and quasi-Monte Carlo methods are used.

Several families of copulas are considered in this study to illustrate the correlation between real estate–particularly hospitality real estate investment trust- and the tourism sector. In essence, this study uses Elliptical copulas and Archimedean copulas, and more recent classes, like extreme value copulas and mixed copulas to conduct the experiment. Under a specific data set, it is revealed that the classical classes of copulas i.e., Elliptical and Archimedean, are selected most often for illustrating the dependency, followed by the extreme value class, particularly the Husler-Reiss copula. However, surprisingly, the mixed copula is not entirely preferable for this data set.

A hidden Markov random field is proposed for the analysis of spatial cylindrical series. The model is a mixture of copula-based bivariate densities, whose parameters vary across space according to a latent random field. It is exploited to segment coastal currents data within a finite number of latent classes that represent specific environmental conditions.

The Farlie-Gumbel-Morgenstern copulas are related to the independence copula \(\varPi \) and can be seen as perturbations of \(\varPi \). Based on quadratic constructions of copulas, we provide a new look at them. Starting from any 2-dimensional copula and an appropriate real function, we introduce new parametric families of copulas which in the case of the independence copula \(\varPi \) coincide with the Farlie-Gumbel-Morgenstern family. Using the proposed approach, we also obtain as a particular case a subclass of the Frechet family of copulas containing all three basic copulas \(W, \varPi \) and M, i.e. a comprehensive family of copulas. Finally, based on an iterative approach, we introduce copula families \(\left( C_r\right) _{r\in [-\infty ,\infty ]}\) complete w.r.t. dependence parameters, resulting in the case of the independence copula and parameters \(r\in [-1,1]\) in the Farlie-Gumbel-Morgenstern family.

This study aims to examine the relationship between oil prices and stock markets in five ASEAN countries: Thailand, Indonesia, Malaysia, Singapore, and the Philippines. Copula approach is used for modelling dependence structure between variables. In essence, this study considers four classes of copula, namely Archimedean copulas, Elliptical copulas, extreme value copulas, and mixed copulas, to examine the dependency between oil prices and stock market prices. We found that Thai, Malaysian, and Indonesian stock markets are likely to boom when crude oil prices increase while the Singaporean stock market as well as the Philippines’s stock market tend to move in the opposite direction to crude oil prices. However, the results show that these relationships are not strong.

In this paper we provide an extension of special parametric class of perturbations of an arbitrary copula (given in [3]) that represent a partial generalization of the FGM family of copulas for parameters from the unit interval. However the FGM family is defined for parameters from the interval \([-1,1]\). We present a construction of perturbations of an arbitrary copula also for parameters from the interval \([-1,0]\) so that together with the former family of perturbations of copulas we get a generalization of the FGM family for the whole interval \([-1,1]\). We also investigated the influence of the parameters of the introduced class of perturbations of copulas on several measures of dependence (Spearman’s rho, Blomqvist’s beta, Gini’s gamma, Kendall’s tau).

We show several interesting examples of connection between distribution of a positively valued random variable and an Archimedean copula through Williamson’s transformation (and Laplace transform), especially when arranged in a sequence. Naturally, there appears a question: how can we use statistical properties of distance functions to draw statistical properties of copulas, and vice versa? This question is formulated in two open problems.

This work shows how dependence in many-valued logic and probability theory can be fused into one concept by using copulas and marginal probabilities. It also shows that the t-norm concept used in fuzzy logic is covered by this approach. This leads to a more general statement that axiomatic probability theory covers logic structure of fuzzy logic. This paper shows the benefits of using structures that go beyond the simple concepts of classical logic and set theory for the modeling of dependences.

In the next few chapters, we study some efficient numerical methods for the valuation of large basket credit derivatives. While the approaches are presented in the Marshall-Olkin copula model, most of the numerical techniques are generic and could be used with other copulas as well. The methods presented span a large spectrum of applied mathematics: Fourier transforms, changes of probability measure, numerical stable schemes, high-dimensional Sobol integration, recursive convolution algorithms.

We review some recent clustering methods based on copulas. Specifically, in the dissimilarity–based clustering framework, we describe and compare methods based on concordance or tail-dependence concept. An illustration is hence provided by using a time series dataset formed by the constituent data of the S&P 500 observed during the financial crisis of 2007-2008. Next, in the likelihood–based clustering framework, we present and discuss a clustering algorithm based on copula and called CoClust. Here, an application to the gene expression profiles of human tumour cell lines is provided to describe the methodology. Finally, a comparison between the two different approaches is performed through a case study on environmental data.

Most common parametric families of copulas are totally ordered, and in many cases they are also positively or negatively regression dependent and therefore they lead to monotone regression functions, which makes them not suitable for dependence relationships that imply or suggest a non-monotone regression function. A gluing copula approach is proposed to decompose the underlying copula into totally ordered copulas that once combined may lead to a non-monotone regression function.

The distinct points of view about factors driving economic growth are introduced all the time in which some effectively useful suggestions then become the growth theories, which in turn lead to various researches on economic growth. This paper aims to examine the joint validity of the growth theories using our introduced model named copula based seemingly unrelated quantile kink regression as a key tool in this work. We concentrate exclusively on the experience of Thailand and found that the growth models can prove their validities for the Thai economy through this experiment.

For data analysis with copulas, one tries to match features seen in data to properties of parametric copula models. Relevant tail asymmetry and dependence properties and measures are summarized. New parametric bivariate copula families in the class of [Durante and Jaworski, 2012] are presented, and some of their dependence and asymmetry properties are determined.

The main problems related to copula sections are reviewed and various methods for constructing copulas that preserve some partial information, are presented. Here, the scope of interest extends from seminal work on the existence and construction of copulas with given diagonal sections to the most recent research on copulas with given diagonal and opposite diagonal sections. Also a survey is given on the state of the art on related domains such as the construction of special families of copulas and generalizations of the copula concept.

We discuss some copula constructions by means of ultramodular bivariate copulas. In general, the ultramodularity of a real function is a stronger version of both its convexity and its supermodularity (the latter property being always satisfied in the case of a bivariate copula). In a statistical sense, ultramodular bivariate copulas are related to random vectors whose components are mutually stochastically decreasing with respect to each other. Analytically speaking, an ultramodular bivariate copula is characterized by the convexity of all of its horizontal and vertical sections. Among other results, we give a sufficient condition for the additive generators of Archimedean ultramodular bivariate copulas, and we propose two constructions for bivariate copulas: the first one being based on ultramodular aggregation functions, and the other one showing the special role of ultramodularity and Schur concavity for a product-like composition of bivariate copulas being again a bivariate copula.

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. First, we make explicit the geometrical notion of symmetry of a function on the unit square with respect to a curve. Next, we provide a measure for quantifying to what extent a quasi-copula or copula can be regarded asymmetric with respect to a given curve. Finally, we derive a lower and upper bound on the degree of asymmetry a quasi-copula can possess with respect to a given curve and show that each bound is sharp within the class of copulas.

We use the concept of copula and extreme value theory to evaluate the impact of extreme events such as flooding, nuclear disaster, etc. on the industry index portfolio. A t copulas based on GARCH model is applied to explain a portfolio risk management with high-dimensional asset allocation. Finally, we calculate the condition Value-at-Risk (CVaR) with the hypothesis of t joint distribution to construct the potential frontier of the portfolio during the times of crisis.

We introduce a class of multivariate non-exchangeable copulas which generalizes many known bivariate FGM type copula families. The properties such as moments, affiliation, association, and positive lower orthant dependent of the proposed class of copula are studied. The simple-to-use multiple regression function and multiple dependence measure formula for this new class of copulas are derived. Several examples are given to illustrate the main results obtained in this paper.

This study explores the calibration of market risk measures during period of economic downturn. This calibration is done in two frameworks: firstly individual profit and loss distribution is modelled using two different types of extreme value distribution namely: the generalized extreme value (GEV) distribution, and the generalized Pareto distribution (GPD). The resulting shape parameters are all positive indicating that these distributions can in fact capture the negative skewness and excess kurtosis of the profit and loss (PL we consider one elliptical copula (student t copula) and one Archimedean copula (Gumbel copula). Using two stock market indices we compute what we refer to as EVT based mark risk measures and the copula based market risk measures for both the left and right tails of the P&L distribution. Our results suggest that copula based risk measures are more reliable in predicting the behavior of market risks during period of economic downturn.

We propose a method to construct a 3-dimensional symmetric function that is radially symmetric, using two symmetric 2-copulas, with one of them being also radially symmetric. We study the properties of the presented construction in some specific cases and provide several examples for different families of copulas.

Recently, fuzzy implications based on copulas, i.e. probabilistic implications and probabilistic S-implications, were introduced and their properties were explored. However, the reverse problem of copulas derived from fuzzy implications, suggested by Massanet et al. [11, 12], is also of interest. In the paper we consider geometric properties of those fuzzy implications that generate copulas. Moreover, we consider the reverse problem for some generalizations of copulas like quasi-copulas and semi-copulas.