Extreme Value Copulas Research Articles

The General Tail Dependence Function in the Marshall-Olkin and Other Parametric Copula Models with an Application to Financial Time Series

The accurate understanding of the dependence structure implied by the parametric models studied in statistical and financial literature has drawn growing attention in recent times. In particular, tail dependence is crucial in this analysis. We study the general tail dependence function in some of the most common copula models found in literature, in which this function has not been obtained. These models are used by statisticians and practitioners alike. With the general tail dependence function, we cover positive and non-positive tail dependence often overlooked. The use of the general tail dependence function generalises the well known approach of using the survival copula to tackle upper tail dependence. We present relevant results regarding tail dependence related functions. We include a broad guide for bivariate families and Hierarchical Archimedean copulas in dimensions 3 and 4. In the multivariate case we study the Marshall-Olkin copula, examples of models based on Max-Id distributions and Extreme Value copulas among others. In an empirical section we exemplify with real data the usefulness of our results by modelling arbitrary types of tail dependence. Furthermore we show how our approach can yield better estimates of Value at Risk than other standard approaches.

Markov product invariance in classes of bivariate copulas characterized by univariate functions

We extend and sharpen some results in the literature concerning the notion of Markov product idempotence in some well-known classes of copulas. Focusing on families of copulas which are characterized by univariate functions we show that in the class of extreme-value copulas, in the class of diagonal copulas and in some special class of copulas represented by measure-preserving transformations only the usual suspects (if contained in the class) are idempotent, namely the product copula Π and minimum copula M. Additionally, we prove a conjecture going back to Albanese and Sempi in 2016 saying that the only idempotent Archimedean copula is the product copula Π.

Conditional normal extreme-value copulas

We propose a new class of extreme-value copulas which are extreme-value limits of conditional normal models. Conditional normal models are generalizations of conditional independence models, where the dependence among observed variables is modeled using one unobserved factor. Conditional on this factor, the distribution of these variables is given by the Gaussian copula. This structure allows one to build flexible and parsimonious models for data with complex dependence structures, such as data with spatial dependence or factor structure. We study the extreme-value limits of these models and show some interesting special cases of the proposed class of copulas. We develop estimation methods for the proposed models and conduct a simulation study to assess the performance of these algorithms. Finally, we apply these copula models to analyze data on monthly wind maxima and stock return minima.

A geometric investigation into the tail dependence of vine copulas

Vine copulas are a type of multivariate dependence model, composed of a collection of bivariate copulas that are combined according to a specific underlying graphical structure. Their flexibility and practicality in moderate and high dimensions have contributed to the popularity of vine copulas, but relatively little attention has been paid to their extremal properties. To address this issue, we present results on the tail dependence properties of some of the most widely studied vine copula classes. We focus our study on the coefficient of tail dependence and the asymptotic shape of the sample cloud, which we calculate using the geometric approach of Nolde (2014). We offer new insights by presenting results for trivariate vine copulas constructed from asymptotically dependent and asymptotically independent bivariate copulas, focusing on bivariate extreme value and inverted extreme value copulas, with additional detail provided for logistic and inverted logistic examples. We also present new theory for a class of higher dimensional vine copulas, constructed from bivariate inverted extreme value copulas.

Modelling High Dimensional Paddy Production Data using Copulas

As the climate change is likely to be adversely affecting the yield of paddy production, thence it has brought a limelight of the probable challenges on human particularly regional food security issues. This paper aims to fit multivariate time series of paddy production variables using copula functions and predicts the next year event based on the data of five countries in southeast Asia. In particular, the most appropriate marginal distribution for each univariate time series was first identified using maximum likelihood parameter estimation method. Next, we performed multivariate copula fitting using two types of copula families, namely, elliptical copula family and Archimedean copula family. Elliptical copula family studied are normal and t copula, while Archimedean copula family considered are Joe, Clayton and Gumbel copulas. The performance of marginal distribution and copula fitting was examined using Akaike information criterion (AIC) values. Finally, we used the best fitted copula model to forecast the succeeding event. In order to assess the performance of copula function, we computed the forecast means and estimation errors of copula function with a generalized autoregressive conditional heteroskedasticity model as reference group. Based on the smallest AIC, the majority of the data favoured the Gumbel copula, which belongs to Archimedean copula family as well as extreme value copula family. Likewise, applying the historical data to forecast the future trends may assist all relevant stakeholders, for instance government, NGO agencies, and professional practitioners in making informed decisions without compromising the environmental as well as economical sustainability in the region.

On the structure of exchangeable extreme-value copulas

We show that the set of d-variate symmetric stable tail dependence functions is a simplex and we determine its extremal boundary. The subset of elements which arises as d-margins of the set of (d+k)-variate symmetric stable tail dependence functions is shown to be proper for arbitrary k≥1. Finally, we derive an intuitive and useful necessary condition for a bivariate extreme-value copula to arise as bi-margin of an exchangeable extreme-value copula of arbitrarily large dimension, and thus to be conditionally iid.

Effect of Seasonality on the Quantiles Estimation of Maximum Floodwater Levels in a Reservoir and Maximum Outflows

Certain relevant variables for dam safety and downstream safety assessments are analyzed using a stochastic approach. In particular, a method to estimate quantiles of maximum outflow in a dam spillway and maximum water level reached in the reservoir during a flood event is presented. The hydrological system analyzed herein is a small mountain catchment in north Spain, whose main river is a tributary of Ebro river. The ancient Foradada dam is located in this catchment. This dam has no gates, so that flood routing operation results from simple consideration of fixed crest spillway hydraulics. In such case, both mentioned variables (maximum outflow and maximum reservoir water level) are basically derived variables that depend on flood hydrograph characteristics and the reservoir’s initial water level. A Monte Carlo approach is performed to generate very large samples of synthetic hydrographs and previous reservoir levels. The use of extreme value copulas allows the ensembles to preserve statistical properties of historical samples and the observed empirical correlations. Apart from the classical approach based on annual periods, the modelling strategy is also applied differentiating two subperiods or seasons (i.e., summer and winter). This allows to quantify the return period distortion introduced when seasonality is ignored in the statistical analysis of the two relevant variables selected for hydrological risk assessment. Results indicate significant deviations for return periods over 125 years. For the analyzed case study, ignoring seasonal statistics and trends, yields to maximum outflows underestimation of 18% for T = 500 years and 29% for T = 1000 years were obtained.

Extra-parametrized extreme value copula : Extension to a spatial framework

Hazard assessment at a regional scale may be performed thanks to a spatial model for maxima that can be obtained by combining the generalized extreme-value (GEV) distribution for the univariate marginal distributions with extreme-value copulas to describe their dependence structure, as justified by the theory of multivariate extreme values. A flexible class of extreme-value copulas, called XGumbel for short, combines two Gumbel copulas with extra-parameters weighting each dimension. In a multisite study, the XGumbel copula quickly becomes over-parametrized. In addition, interpolation to ungauged locations is not easily achieved. We develop an extension of the XGumbel copula to the spatial framework by defining the extra-parameters as a mapping shaped as a disk. The inference of the Spatialized XGumbel copula is performed thanks to an Approximate Bayesian Computation (ABC) scheme with summary statistics based on upper tail dependence coefficients. The GEV parameters are estimated with a spatial regression model built with a vector generalized linear model. We evaluate and compare this spatial model with the Brown–Resnick process on annual maxima of daily precipitation totals at 177 gauged stations in the French Mediterranean over a 57 year period. Our analyses show that the ABC scheme yields, except in one instance, interpretable parameters. In addition, the Spatialized XGumbel copula is able to reproduce reasonably well the non-stationarity present in our case study.

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

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.

Modeling the Dependency between Extreme Prices of Selected Agricultural Products on the Derivatives Market Using the Linkage Function

The purpose of the article is to identify and estimate the dependency model for the extreme prices of agricultural products listed on the Chicago Mercantile Exchange. The article presents the results of the first stage of research covering the time interval 1975–2010. The selected products are: Corn, soybean and wheat. The analysis of the dependency between extreme price values on the selected futures was based on the estimation of five models of two-dimensional extreme value copulas, namely, the Galambos copula, the Gumbel copula, the Husler–Reiss copula, the Tawn asymmetric copula and the t-EV copula. The next stage of the analysis was to test whether the structure of the dependency described with the estimated copulas is a sufficient approximation of reality, and whether it is suitable for modeling empirical data. The quality of matching the estimated copulas to empirical data of return rates of agricultural products was assessed. For this purpose, the Kendall coefficient was calculated, and the methodology of the empirical combining function was used. The conducted research allowed for the determination of the conduct for this kind of phenomena as it is crucial in the process of investing in derivatives markets. The analyzed phenomena are highly dependent on e.g., financial crises, war, or market speculation but also on drought, fires, rainfall, or even crop oversupply. The conducted analysis is of key importance in terms of balancing agricultural production on a global scale. It should be emphasized that conducting market analysis of agricultural products at the Chicago Mercantile Exchange in the context of competition with the agricultural market of the European Union is of significant importance.

Extreme biconic copulas: Characterization, properties and extensions to aggregation functions

Motivated by optimization problems arising in the study of copula-based stochastic models, we study the extreme points (in the Krein–Milman sense) of the class of extreme biconic copulas and provide their characterization. Moreover, we show how this characterization may help in shedding light upon new aspects of biconic copulas, including their link with extreme-value copulas. The study is embedded into the more general setting of aggregation functions (especially, semi–copulas and quasi–copulas), where the determination of extreme points has also interest.

Nonparametric dependence modeling via cluster analysis: A financial contagion application

Dependence measures, from linear correlation coefficients to recent copula-based methods, have been widely used to find out associations between variables. Although the latter type of measure has overcome many drawbacks of traditional measures, copula has intrinsically some undesirable characteristics for particular applications. In this paper, we discuss dependence modeling from a pattern recognition perspective and then introduce a new non-parametric approach based on anomaly detection through cluster analysis. The proposed methodology uses a weighting procedure based on Voronoi cells densities, named Weighted Voronoi Distance (WVD), to identify potentially atypical associations between univariate time series. The advantages are two-fold. First, the time series structure is respected and neither independence nor homoscedasticity is presumed within data. Second, any distribution of the data and any dependence function is allowed. An inference procedure is presented and simulation studies help to visualize the behavior and benefits of the proposed measure. Finally, real financial data is used to analyze the detection capacity of the contagion effect in financial markets during the 2007 sub-prime crisis. Different asset classes were included, and the WVD was able to signalize anomalies more strongly than the Extreme Value Theory and copula approach.

Subordinators which are infinitely divisible w.r.t. time: Construction, properties, and simulation of max-stable sequences and infinitely divisible laws

The concept of a L\'evy subordinator is generalized to a family of non-decreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the L\'evy subordinator special case, the considered family is always strongly infinitely divisible with respect to time, meaning that a path can be represented in distribution as a finite sum with arbitrarily many summands of independent and identically distributed paths of another process. Besides distributional properties of the process, we present two applications to the design of accurate and efficient simulation algorithms. First, each member of the considered family corresponds uniquely to an exchangeable max-stable sequence of random variables, and we demonstrate how the associated extreme-value copula can be simulated exactly and efficiently from its Pickands dependence measure. Second, we show how one obtains different series and integral representations for infinitely divisible probability laws by varying the parameterizing pair of Bernstein functions, without changing the law of one-dimensional margins of the process. As a particular example, we present an exact simulation algorithm for compound Poisson distributions from the Bondesson class, for which the generalized inverse of the distribution function of the associated Stieltjes measure can be evaluated accurately.

A sharp inequality for Kendall’s τ and Spearman’s ρ of Extreme-Value Copulas

AbstractWe derive a new (lower) inequality between Kendall’s τ and Spearman’s ρ for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.

On the size of the class of bivariate extreme-value copulas with a fixed value of Spearman's rho or Kendall's tau

Abstract A bivariate extreme-value copula is characterized by a function of one variable, called a Pickands dependence function, which is convex and comprised between two bounds. The authors identify the smallest possible compact set containing the graph of all Pickands dependence functions whose corresponding bivariate extreme-value copula has a fixed value of Spearman's rho or Kendall's tau. The consequences of this result for statistical modeling are outlined.

Synthetic Hydrographs Generation Downstream of a River Junction Using a Copula Approach for Hydrological Risk Assessment in Large Dams

Peak flows values (Q) and hydrograph volumes (V) are obtained from a selected family of historical flood events (period 1957–2017), for two neighboring mountain catchments located in the Ebro river basin, Spain: rivers Ésera and Isábena. Barasona dam is located downstream of the river junction. The peaks over threshold (POT) method is used for a univariate frequency analysis performed for both variables, Q and V, comparing several suitable distribution functions. Extreme value copulas families have been applied to model the bivariate distribution (Q, V) for each of the rivers. Several goodness-of-fit tests were used to assess the applicability of the selected copulas. A similar copula approach was carried out to model the dependence between peak flows of both rivers. Based on the above-mentioned statistical analysis, a Monte Carlo simulation of synthetic design flood hydrographs (DFH) downstream of the river junction is performed. A gamma-type theoretical pattern is assumed for partial hydrographs. The resulting synthetic hydrographs at the Barasona reservoir are finally obtained accounting for flow peak time lag, also described in statistical terms. A 50,000 hydrographs ensemble was generated, preserving statistical properties of marginal distributions as well as statistical dependence between variables. The proposed method provides an efficient and practical modeling framework for the hydrological risk assessment of the dam, improving the basis for the optimal management of such infrastructure.

Modelling the Dependence Structure of Financial Assets: A Bivariate Extreme Data Study

Quantifying dependence among variables is the core of all modelling efforts in financial econometrics. In the recent years, copula was introduced to model the dependence structure among financial assets return, and its application developed fast. A large number of studies on copula have been performed, but the study of multivariate extremes related with copulas was quite behind in comparison with the research on copulas. In order to predict extreme losses in finance, extreme value copulas play more important role than copulas. In this paper, we study the modelling of extreme value dependence using extreme value copulas on finance data and introducing a negative dependence between the assets return. This model was applied in the portfolio of the IDX Composite Index (IHSG) and the exchange rate for Indonesia Rupiah to United States Dollar (NT). Each individual asset return is modelled by the AR-GARCH and the joint distribution is modelled using extreme value copulas. As a comparison, the joint distribution is also modelled using the copulas. The result shows that the dependence between IHSG and NT is negative. Therefore, in this paper, we tested the extreme value copulas and the copulas which were rotated 270 degree and 90 degree. The empirical study showed that the extreme value copulas rotated with 270 degree are relatively more appropriate model than the copulas. Quantifying dependence among variables is the core of all modelling efforts in financial econometrics. In the recent years, copula was introduced to model the dependence structure among financial assets return, and its application developed fast. A large number of studies on copula have been performed, but the study of multivariate extremes related with copulas was quite behind in comparison with the research on copulas. In order to predict extreme losses in finance, extreme value copulas play more important role than copulas. In this paper, we study the modelling of extreme value dependence using extreme value copulas on finance data and introducing a negative dependence between the assets return. This model was applied in the portfolio of the IDX Composite Index (IHSG) and the exchange rate for Indonesia Rupiah to United States Dollar (NT). Each individual asset return is modelled by the AR-GARCH and the joint distribution is modelled using extreme value copulas. As a comparison, the joint distribution is also modelled using the copulas. The result shows that the dependence between IHSG and NT is negative. Therefore, in this paper, we tested the extreme value copulas and the copulas which were rotated 270 degree and 90 degree. The empirical study showed that the extreme value copulas rotated with 270 degree are relatively more appropriate model than the copulas.

Spatial Cluster Detection in Mobility Networks: A Copula Approach

Summary In mobility network capacity planning, characterizing the mobility network traffic is one of the most challenging tasks. Besides the growth trend and multiple periodic temporal patterns for normal traffic, the problem is complicated by the occasionally intense traffic for special events and its dynamic spatial relationships. Identifying the areas that have different traffic patterns compared with their neighbouring areas is a problem of spatial hotspot detection. In the paper, a copula-based method is proposed: using a multivariate extreme value copula, the upper tail dependence of the traffic distributions of neighbouring cell towers is evaluated, and then a cluster of multiple time series (i.e. multiple cell towers) with high upper tail dependence is detected. The method proposed is validated by using synthetic data as well as real mobility traffic data.

Assessing the copula selection for bivariate frequency analysis based on the tail dependence test

The flood characteristics, namely, peak, duration and volume provide important information for the design of hydraulic structures, water resources planning, reservoir management and flood hazard mapping. Flood is a complex phenomenon defined by strongly correlated characteristics such as peak, duration and volume. Therefore, it is necessary to study the simultaneous, multivariate, probabilistic behaviour of flood characteristics. Traditional multivariate parametric distributions have widely been applied for hydrological applications. However, this approach has some drawbacks such as the dependence structure between the variables, which depends on the marginal distributions or the flood variables that have the same type of marginal distributions. Copulas are applied to overcome the restriction of traditional bivariate frequency analysis by choosing the marginals from different families of the probability distribution for flood variables. The most important step in the modelling process using copula is the selection of copula function which is the best fit for the data sample. The choice of copula may significantly impact the bivariate quantiles. Indeed, this study indicates that there is a huge difference in the joint return period estimation using the families of extreme value copulas and no upper tail copulas (Frank, Clayton and Gaussian) if there exists asymptotic dependence in the flood characteristics. This study suggests that the copula function should be selected based on the dependence structure of the variables. From the results, it is observed that the result from tail dependence test is very useful in selecting the appropriate copula for modelling the joint dependence structure of flood variables. The extreme value copulas with upper tail dependence have proved that they are appropriate models for the dependence structure of the flood characteristics and Frank, Clayton and Gaussian copulas are the appropriate copula models in case of variables which are diagnosed as asymptotic independence.

On the study of extremes with dependent random right-censoring

The study of extremes in missing data frameworks is a recent developing field. In particular, the randomly right-censored case has been receiving a fair amount of attention in the last decade. All studies on this topic, however, essentially work under the usual assumption that the variable of interest and the censoring variable are independent. Furthermore, a frequent characteristic of estimation procedures developed so far is their crucial reliance on particular properties of the asymptotic behaviour of the response variable Z (that is, the minimum between time-to-event and time-to-censoring) and of the probability of censoring in the right tail of Z. In this paper, we focus instead on elucidating this asymptotic behaviour in the dependent censoring case, and, more precisely, when the structure of the dependent censoring mechanism is given by an extreme value copula. We then draw a number of consequences of our results, related to the asymptotic behaviour, in this dependent context, of a number of estimators of the extreme value index of the random variable of interest that were introduced in the literature under the assumption of independent censoring, and we discuss more generally the implications of our results on the inference of the extremes of this variable.