Extreme Value Copulas Research Articles

Exact simulation of reciprocal Archimedean copulas

The decreasing enumeration of the points of a Poisson random measure whose mean measure is Radon on (0,∞] can be represented as a non-increasing function of the jump times of a standard Poisson process. This observation allows to generalize the essential idea from a well-known exact simulation algorithm for arbitrary extreme-value copulas to copulas of more general max-infinitely divisible distributions, with reciprocal Archimedean copulas being a particular example.

Retracted: Understanding the relationship of tourism demands connecting with economy and tourism stock index in an extreme case of Thailand bivariate extreme value copula approach

Interestingly, there is no confirmation that the Bivariate Extreme Value models and Bivariate Extreme Value Copula models are employed to explore tourism variables, especially studying in Thailand. This paper aims to statistically clarify extreme cases in tourism demands connecting with a macroeconomic level and micro tourism factor. Three quarterly indexes, including Thailand economic expansion rates, CENTEL stock indexes, and tourist arrivals were collected during 2003 to 2015. Empirically, the results showed extreme value copula models can better capture the dependence structure between three economic components than using the bivariate extreme value models. In addition, the extreme dependent results among the three comparable variables can be implied that the special tourism policies for extreme situations should not be neglected. It is reasonable to simultaneously adjusted tourism policies to both normal cases and critical situations during the whole year.

Extreme-value copulas associated with the expected scaled maximum of independent random variables

It is well-known that the expected scaled maximum of non-negative random variables with unit mean defines a stable tail dependence function associated with some extreme-value copula. In the special case when these random variables are independent and identically distributed, min-stable multivariate exponential random vectors with the associated survival extreme-value copulas are shown to arise as finite-dimensional margins of an infinite exchangeable sequence in the sense of De Finetti’s Theorem. The associated latent factor is a stochastic process which is strongly infinitely divisible with respect to time, which induces a bijection from the set of distribution functions F of non-negative random variables with finite mean to the set of Lévy measures ν on (0,∞]. Since the Gumbel and the Galambos copula are the most popular examples of this construction, the investigation of this bijection contributes to a further understanding of their well-known analytical similarities. Furthermore, a simulation algorithm based on the latent factor representation is developed, if the support of F is bounded. Especially in large dimensions, this algorithm is efficient because it makes use of the De Finetti structure.

Weak convergence of the weighted empirical beta copula process

The empirical copula has proved to be useful in the construction and understanding of many statistical procedures related to dependence within random vectors. The empirical beta copula is a smoothed version of the empirical copula that enjoys better finite-sample properties. At the core lie fundamental results on the weak convergence of the empirical copula and empirical beta copula processes. Their scope of application can be increased by considering weighted versions of these processes. In this paper we show weak convergence for the weighted empirical beta copula process. The weak convergence result for the weighted empirical beta copula process is stronger than the one for the empirical copula and its use is more straightforward. The simplicity of its application is illustrated for weighted Cramér–von Mises tests for independence and for the estimation of the Pickands dependence function of an extreme-value copula.

From Weakly Chaotic Dynamics to Deterministic Subdiffusion via Copula Modeling

Copula modeling consists in finding a probabilistic distribution, called copula, whereby its coupling with the marginal distributions of a set of random variables produces their joint distribution. The present work aims to use this technique to connect the statistical distributions of weakly chaotic dynamics and deterministic subdiffusion. More precisely, we decompose the jumps distribution of Geisel-Thomae map into a bivariate one and determine the marginal and copula distributions respectively by infinite ergodic theory and statistical inference techniques. We verify therefore that the characteristic tail distribution of subdiffusion is an extreme value copula coupling Mittag-Leffler distributions. We also present a method to calculate the exact copula and joint distributions in the case where weakly chaotic dynamics and deterministic subdiffusion statistical distributions are already known. Numerical simulations and consistency with the dynamical aspects of the map support our results.

Copula model dependency between oil prices and stock markets: evidence from Tunisia and Egypt

In this work, our objective is to study in a first step the relationships between oil price and stock market indices in Tunisia and Egypt using copulas, and then in a second step to analyse the optimal weights and hedge ratio for building optimal portfolios to minimise the exposure to risk from oil price changes. The model is implemented with an ARMA–GARCH–GPD using daily observations for the 2 January 1998 to 31 December 2013 period for the marginal distribution and the extreme value copula for the joint distribution, which allows taking into account non-linear dependence, tails behaviour and their development over time. Various copula functions are used to model the dependence structure between oil prices and stock markets of Tunisia and Egypt. Empirical results provide evidence of significant and symmetric tail dependence for international oil prices and stock markets. On the other hand, volatility for this stock markets registered record levels due to the increase of the degree of oil prices. However, dependence structure changed over time, indicating that hedging effectiveness and hedge ratio for the oil asset in the hedged portfolios varied over time.

Tail-weighted dependence measures with limit being the tail dependence coefficient

ABSTRACTFor bivariate continuous data, measures of monotonic dependence are based on the rank transformations of the two variables. For bivariate extreme value copulas, there is a family of estimators , for , of the extremal coefficient, based on a transform of the absolute difference of the α power of the ranks. In the case of general bivariate copulas, we obtain the probability limit of as the sample size goes to infinity and show that (i) for is a measure of central dependence with properties similar to Kendall's tau and Spearman's rank correlation, (ii) is a tail-weighted dependence measure for large α, and (iii) the limit as is the upper tail dependence coefficient. We obtain asymptotic properties for the rank-based measure and estimate tail dependence coefficients through extrapolation on . A data example illustrates the use of the new dependence measures for tail inference.

A general approach to full-range tail dependence copulas

Full-range tail dependence copulas have recently been proved very useful for modeling various dependence patterns in the joint distributional tails. However, there are only a few applicable candidate models that have the full-range tail dependence property. In this paper, we present a general approach to constructing bivariate copulas that have full-range tail dependence in both upper and lower tails and are able to account for both reflection symmetry and reflection asymmetry. The general approach is based on mixtures of positive regularly varying random variables, and the full-range tail dependence property is established for such a general model. In order to construct copulas that possess the above dependence properties and are fast to compute, we construct a full-range tail dependence copula based on mixtures of Pareto random variables. We derive dependence properties of the proposed copula, and the extreme value copula based on it. A comparison with the full-range tail dependence copula proposed in Hua (2017) has been conducted, and the computational speed has been largely improved by the copula proposed in the current paper.

Bivariate nonparametric estimation of the Pickands dependence function using Bernstein copula with kernel regression approach

In this study, a new nonparametric approach using Bernstein copula approximation is proposed to estimate Pickands dependence function. New data points obtained with Bernstein copula approximation serve to estimate the unknown Pickands dependence function. Kernel regression method is then used to derive an intrinsic estimator satisfying the convexity. Some extreme-value copula models are used to measure the performance of the estimator by a comprehensive simulation study. Also, a real-data example is illustrated. The proposed Pickands estimator provides a flexible way to have a better fit and has a better performance than the conventional estimators.

AN EVENT-BASED APPROACH FOR EXTREME JOINT PROBABILITIES OF WAVES AND SEA LEVELS

A methodology for determining extreme joint probabilities of two metocean variables, in particular wave height and sea level, is presented in the paper. This methodology focuses in particular on the sampling of the time series, which should be based on the notion of event: either the event generating the variables whose joint probabilities are wanted (such as a storm generating waves and surges) or the event that is a result of the combination of these variables (such as a beach erosion event generated by waves at high sea level). A classification is proposed for multivariate analyses in order to help the choice of the sampling method. The dependence between the variables is analysed using tools such as the chi-plot, of which an enhanced presentation is proposed, then is modelled by extreme-value copulas (Gumbel-Hougaard, Galambos and Hüsler-Reiss) estimated by Canonical Maximum Likelihood or by the upper tail dependence coefficient. Joint return periods are then computed. A comparison is made with a simulation from the JOIN-SEA software on a dataset of wave height and sea levels offshore Brest, France. Then the bivariate methodology is extended to a multivariate framework. The distribution of sea level is determined by an indirect approach (extrapolation of extreme surges then convolution with the astronomical tide) and the dependence is analyzed between the wave height and the surge component only. A bidimensional convolution between the joint distribution of wave height and surge and the distribution of the astronomical tide yields the joint distribution of wave height and sea level. The application of this method to the dataset of Brest and its comparison with the bivariate approach are finally discussed.

Extremal attractors of Liouville copulas

Liouville copulas, which were introduced in McNeil and Neslehova (2010), are asymmetric generalizations of the ubiquitous Archimedean copula class. They are the dependence structures of scale mixtures of Dirichlet distributions, also called Liouville distributions. In this paper, the limiting extreme-value copulas of Liouville copulas and of their survival counterparts are derived. The limiting max-stable models, termed here the scaled extremal Dirichlet, are new and encompass several existing classes of multivariate max-stable distributions, including the logistic, negative logistic and extremal Dirichlet. As shown herein, the stable tail dependence function and angular density of the scaled extremal Dirichlet model have a tractable form, which in turn leads to a simple de Haan representation. The latter is used to design efficient algorithms for unconditional simulation based on the work of Dombry, Engelke and Oesting (2015) and to derive tractable formulas for maximum-likelihood inference. The scaled extremal Dirichlet model is illustrated on river flow data of the river Isar in southern Germany.

Multivariate extreme value copulas with factor and tree dependence structures

Parsimonious extreme value copula models with O(d) parameters for d observed variables of extrema are presented. These models utilize the dependence characteristics, including factor and tree structures, assumed on the underlying variables that give rise to the data of extremes. For factor structures, a class of parametric models is obtained by taking the extreme value limit of factor copulas with non-zero tail dependence. An alternative model suitable for both factor and tree structures imposes constraints on the parametric Husler-Reiss copula to get representations in terms of O(d) other parameters. Dependence properties are discussed. As the full density is often intractable, the method of composite (pairwise) likelihood is used for model inference. Procedures to improve the stability of bivariate density evaluation are also developed. The proposed models are applied to two data examples — one for annual extreme river flows and one for bimonthly extremes of daily stock returns.

A new estimator for the Pickands dependence function

The Pickands dependence function characterizes an extreme value copula, a useful tool in the modeling of multivariate extremes. A new estimator is presented along with its convergence properties and performance through simulation.

Extraction dependence structure of distorted copulas via a measure of dependence

Copulas are one of the most powerful tools in modeling dependence structure of multivariate variables. In Tran et al. (Integrated uncertainty in knowledge modelling and decision making. Springer, Berlin, pp 126–137, 2015), we have constructed a new measure of dependence, $$\lambda (C),$$ based on Sobolev norm for copula C which can be used to characterize comonotonicity, countermonotonicity and independence of random vectors. This paper aims to use the measure $$ \lambda (C) $$ to study how dependence structure of a distorted copula after being transformed by a distortion function is changed. Firstly, we propose two methods to estimate the measure $$\lambda (C)$$ , one for known copula C using conditional copula-based Monte Carlo simulation and the latter for unknown copula dealing with empirical data. Thereafter, PH-transform $$g_{ PH }$$ of extreme value copulas and Wang’s transform $$ g_\gamma $$ of normal and product copula are studied, and we observe their dependence behaviors changing through variability of the measure $$ \lambda (C) $$ . Our results show that dependence structure of distorted copulas is subject to comonotonicity as increasing the parametric $$ \gamma $$ .

The combined risk of extreme tropical cyclone winds and storm surges along the U.S. Gulf of Mexico Coast

AbstractTropical cyclones, with their nearshore high wind speeds and deep storm surges, frequently strike the United States Gulf of Mexico coastline influencing millions of people and disrupting offshore economic activities. The combined risk of occurrence of tropical cyclone nearshore wind speeds and storm surges is assessed at 22 coastal cities throughout the United States Gulf of Mexico. The models used are extreme value copulas fitted with margins defined by the generalized Pareto distribution or combinations of Weibull, gamma, lognormal, or normal distributions. The statistical relationships between the nearshore wind speed and storm surge are provided for each coastal city prior to the copula model runs using Spearman's rank correlations. The strongest significant relationship between the nearshore wind speed and storm surge exists at Shell Beach, LA (ρ = 0.67), followed by South Padre Island, TX (ρ = 0.64). The extreme value Archimedean copula models for each city then provide return periods for specific nearshore wind speed and storm surge pairs. Of the 22 cities considered, Bay St. Louis, MS, has the shortest return period for a tropical cyclone with at least a 50 ms−1 nearshore wind speed and a 3 m surge (19.5 years, 17.1‐23.5). The 90% confidence intervals are created by recalculating the return periods for a fixed set of wind speeds and surge levels using 100 samples of the model parameters. The results of this study can be utilized by policy managers and government officials concerned with coastal populations and economic activity in the Gulf of Mexico.

An event-based approach for extreme joint probabilities of waves and sea levels

A methodology for determining extreme joint probabilities of two metocean variables, in particular wave height and sea level, is presented in the paper. This methodology focuses in particular on the sampling of the time series, which should be based on the notion of event: either the event generating the variables whose joint probabilities are wanted (such as a storm generating waves and surges) or the event that is a result of the combination of these variables (such as a beach erosion event generated by waves at high sea level). A classification is proposed for multivariate analyses in order to help the choice of the sampling method. The dependence between the variables is analysed using tools such as the chi-plot, of which an enhanced presentation is proposed, then is modelled by extreme-value copulas (Gumbel-Hougaard, Galambos and Hüsler-Reiss) estimated by Canonical Maximum Likelihood or by the upper tail dependence coefficient. Joint return periods are then computed. A comparison is made with a simulation from the JOIN-SEA software on a dataset of wave height and sea levels offshore Brest, France. Then the bivariate methodology is extended to a multivariate framework. The distribution of sea level is determined by an indirect approach (extrapolation of extreme surges then convolution with the astronomical tide) and the dependence is analysed between the wave height and the surge component only. A bidimensional convolution between the joint distribution of wave height and surge and the distribution of the astronomical tide yields the joint distribution of wave height and sea level. The application of this method to the dataset of Brest and its comparison with the bivariate approach are finally discussed.

On a bivariate copula with both upper and lower full-range tail dependence

Copula functions can be useful in accounting for various dependence patterns appearing in joint tails of data. We propose a new two-parameter bivariate copula family that possesses the following features. First, both upper and lower tails are able to explain full-range tail dependence. That is, the dependence in each tail can range among quadrant tail independence, intermediate tail dependence, and usual tail dependence. Second, it can capture upper and lower tail dependence patterns that are either the same or different. We first prove the full-range tail dependence property, and then we obtain the corresponding extreme value copula. There are two applications based on the proposed copula. The first one is modeling pairwise dependence between financial markets. The second one is modeling dynamic tail dependence patterns that appear in upper and lower tails of a loss-and-expense data.

Dependence between Extreme Risk Factor Movements of Euro Area Sovereign Bonds

This paper is the first which studies extreme risk factor movements of Euro Area sovereign bonds, whose prices are largely determined by risk factors representing interest rate risk, credit risk and market liquidity risk. Starting model independently from fundamental no-arbitrage relationships, we disentangle the government bonds' yields into a risk-free rate, the government's credit spread as well as its country - or even instrument-specific market liquidity spread for bonds with five years to maturity. At first, univariate extreme value theory (EVT) is applied to each member state individually to examine the distributional characteristics of extreme risk factor returns (tail risk) during the recent financial market crises between 2008 and 2016. Furthermore, the paper analyzes multivariate risk factor dependencies for each member state. We calculate joint exceedance probabilities for high quantiles for each of the three bivariate risk factor combinations and estimate dependence parameters of various, mainly extreme-value copulas for increasing thresholds to study tail dependence. Finally, we test the goodness-of-fit of the estimated parametric copulas and conclude which copula works best in modelling extremal dependence between interest rate risk, credit risk, and market liquidity risk factor changes during severe financial market turmoil.

Characterizations of bivariate conic, extreme value, and Archimax copulas

Abstract Based on a general construction method by means of bivariate ultramodular copulas we construct, for particular settings, special bivariate conic, extreme value, and Archimax copulas. We also show that the sets of copulas obtained in this way are dense in the sets of all conic, extreme value, and Archimax copulas, respectively.

Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials

Many applications in risk analysis require the estimation of the dependence among multivariate maxima, especially in environmental sciences. Such dependence can be described by the Pickands dependence function of the underlying extreme-value copula. Here, a nonparametric estimator is constructed as the sample equivalent of a multivariate extension of the madogram. Shape constraints on the family of Pickands dependence functions are taken into account by means of a representation in terms of Bernstein polynomials. The large-sample theory of the estimator is developed and its finite-sample performance is evaluated with a simulation study. The approach is illustrated with a dataset of weekly maxima of hourly rainfall in France recorded from 1993 to 2011 at various weather stations all over the country. The stations are grouped into clusters of seven stations, where our interest is in the extremal dependence within each cluster.