We show that all multivariate Extreme Value distributions, which are the possible weak limits of the $K$ largest order statistics of iid sequences, have the same copula, the so called K-extremal copula. This copula is described through exact expressions for its density and distribution functions. We also study measures of dependence, we obtain a weak convergence result and we propose a simulation algorithm for the K-extremal copula.
This paper reviews generalized Pareto copulas (GPC), which turn out to be a key to multivariate extreme value theory. Any GPC can be represented in an easy analytic way using a particular type of norm on $\mathbb{R}^d$, called $D$-norm. The characteristic property of a GPC is its exceedance stability. GPC might help to end the debate: What is a multivariate generalized Pareto distribution? We present an easy way how to simulate data from an arbitrary GPC and, thus, from an arbitrary generalized Pareto distribution. As an application we derive nonparametric estimates of the probability that a r…
The univariate piecing-together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. We propose a multivariate extension. First it is shown that an arbitrary copula is in the domain of attraction of a multivariate extreme value distribution if and only if its upper tail can be approximated by the upper tail of a multivariate GPD with uniform margins. The multivariate PT then consists of two steps: The upper tail of a given copula $C$ is cut off and substituted by a multivariate GPD copula in a continuou…
The core of the classical block maxima method consists of fitting an extreme value distribution to a sample of maxima over blocks extracted from an underlying series. In asymptotic theory, it is usually postulated that the block maxima are an independent random sample of an extreme value distribution. In practice however, block sizes are finite, so that the extreme value postulate will only hold approximately. A more accurate asymptotic framework is that of a triangular array of block maxima, the block size depending on the size of the underlying sample in such a way that both the block size a…
As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables $T_{1},...,T_{r}$. In any case, we assume that $T_{1},...,T_{r}$ are identically distributed, with a common survival function $\overline{G}$ and their survival copula is denoted by $K$. The diagonal's and subdiagonals' sections of $K$, along with $\overline{G}$, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the a…
The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are all…
Under a mild condition we give closed-form expressions for copulas of systems that consist of maxima and of minima of subvectors of a given random vector $X$ with continuous marginals. Said expressions appear explicit in the copula of $X$ and the mentioned condition is for example met when the law of $X$ admits a strictly positive density with respect to Lebesgue measure. In the i.i.d. case these "maxmin" copulae become universal and the conditions on their validity can be dropped entirely. Our main motivation comes from applications to shock models that arise in multivariate survival theory. …
In the field of finance, insurance, and system reliability, etc., it is often of interest to measure the dependence among variables by modeling a multivariate distribution using a copula. The copula models with parametric assumptions are easy to estimate but can be highly biased when such assumptions are false, while the empirical copulas are non-smooth and often not genuine copula making the inference about dependence challenging in practice. As a compromise, the empirical Bernstein copula provides a smooth estimator but the estimation of tuning parameters remains elusive. In this paper, by u…
For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula, uniformly over…
Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric co…
Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance - in fact, given samples of a copula $C$ the corresponding empirical copulas do not converge weakly conditional to $C$ with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and …
Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, e.g. a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of fl…
We offer a new perspective on risk aggregation with FGM copulas. Along the way, we discover new results and revisit existing ones, providing simpler formulas than one can find in the existing literature. This paper builds on two novel representations of FGM copulas based on symmetric multivariate Bernoulli distributions and order statistics. First, we detail families of multivariate distributions with closed-form solutions for the cumulative distribution function or moments of the aggregate random variables. We order aggregate random variables under the convex order and provide methods to comp…
We study the weak convergence of conditional empirical copula processes, when the conditioning event has a nonzero probability. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general - possibly conditional - multivariate dependence measures and their estimators. By applying our theoretical results, we prove the asymptotic normality of some estimators of such dependence measures.
Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estima…
A new class of copulas based on order statistics was introduced by Baker (2008). Here, further properties of the bivariate and multivariate copulas are described, such as that of likelihood ratio dominance (LRD), and further bivariate copulas are introduced that generalize the earlier work. One of the new copulas is an integral of a product of Bessel functions of imaginary argument, and can attain the Fréchet bound. The use of these copulas for fitting data is described, and illustrated with examples. It was found empirically that the multivariate copulas previously proposed are not flexible e…
Our article addresses the problem of flexibly estimating a multivariate density while also attempting to estimate its marginals correctly. We do so by proposing two new estimators that try to capture the best features of mixture of normals and copula estimators while avoiding some of their weaknesses. The first estimator we propose is a mixture of normals copula model that is a flexible alternative to parametric copula models such as the normal and t copula. The second is a marginally adapted mixture of normals estimator that improves on the standard mixture of normals by using information con…
For the analysis of clustered survival data, two different types of models that take the association into account, are commonly used: frailty models and copula models. Frailty models assume that conditional on a frailty term for each cluster, the hazard functions of individuals within that cluster are independent. These unknown frailty terms with their imposed distribution are used to express the association between the different individuals in a cluster. Copula models on the other hand assume that the joint survival function of the individuals within a cluster is given by a copula function, e…
When facing multivariate covariates, general semiparametric regression techniques come at hand to propose flexible models that are unexposed to the curse of dimensionality. In this work a semiparametric copula-based estimator for conditional quantiles is investigated for complete or right-censored data. In spirit, the methodology is extending the recent work of Noh et al. (2013) and Noh et al. (2015), as the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to…
A pair-copula construction is a decomposition of a multivariate copula into a structured system, called regular vine, of bivariate copulae or pair-copulae. The standard practice is to model these pair-copulae parametrically, which comes at the cost of a large model risk, with errors propagating throughout the vine structure. The empirical pair-copula proposed in the paper provides a nonparametric alternative still achieving the parametric convergence rate. It can be used as a basis for inference on dependence measures, for selecting and pruning the vine structure, and for hypothesis tests conc…
We generalize the notion of essential closures which is used in formulating a geometric necessary condition for a set to be the support of a multivariate copula. Furthermore, in some special cases, we derive an explicit formula of the support in terms of essential closures and obtain a stronger necessary condition.
In this paper we review Bernstein and grid-type copulas for arbitrary dimensions and general grid resolutions in connection with discrete random vectors possessing uniform margins. We further suggest a pragmatic way to fit the dependence structure of multivariate data to Bernstein copulas via grid-type copulas and empirical contingency tables. Finally, we discuss a Monte Carlo study for the simulation and PML estimation for aggregate dependent losses form observed windstorm and flooding data.
An extension of the empirical copula is considered by combining an estimator of a multivariate cumulative distribution function with estimators of the marginal cumulative distribution functions for marginal estimators that are not necessarily equal to the margins of the joint estimator. Such a hybrid estimator may be reasonable when there is additional information available for some margins in the form of additional data or stronger modelling assumptions. A functional central limit theorem is established and some examples are developed.
The univariate Piecing-Together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. A multivariate extension was established by Aulbach et al. (2012a): The upper tail of a given copula C is cut off and replaced by a multivariate GPD-copula in a continuous manner, yielding a new copula called a PT-copula. Then each margin of this PT-copula is transformed by a given univariate distribution function. This provides a multivariate distribution function with prescribed margins, whose copula is a GPD-copula t…
This paper deals with dependence across marginally exponentially distributed arrival times, such as default times in financial modeling or inter-failure times in reliability theory. We explore the relationship between dependence and the possibility to sample final multivariate survival in a long time-interval as a sequence of iterations of local multivariate survivals along a partition of the total time interval. We find that this is possible under a form of multivariate lack of memory that is linked to a property of the survival times copula. This property defines a "self-chaining-copula", an…
This paper introduces vector copulas associated with multivariate distributions with given multivariate marginals, based on the theory of measure transportation, and establishes a vector version of Sklar's theorem. The latter provides a theoretical justification for the use of vector copulas to characterize nonlinear or rank dependence between a finite number of random vectors (robust to within vector dependence), and to construct multivariate distributions with any given non overlapping multivariate marginals. We construct Elliptical and Kendall families of vector copulas, derive their densit…
This paper provides bifactor gamma distribution, trivariate gamma distribution and two copula families on [0, 1] n obtained from the Laplace transforms of the multivariate gamma distribution and the multi-factor gamma distribution given by [P ($θ$)] --$λ$ and [P ($θ$)] --$λ$ n i=1 (1 + pi$θ$i) --($λ$ i --$λ$) respectively, where P is an affine polynomial with respect to the n variables $θ$1,. .. , $θ$n. These copulas are new generalized Farlie-Gumbel-Morgenstern copulas and allow in particular to obtain multivariate gamma distributions for which the cumulative distribution functions and the pr…
Building upon earlier work in which axioms were formulated for multivariate measures of concordance, we examine properties of such measures. In particular, we examine the relations between the measure of concordance of an $n$-copula and the measures of concordance of the copula's marginals.
This article continues our study of Markovian consistency and Markov copulae. In particular, we characterize the weak Markovian consistency for finite Markov chains. We discuss some aspects of dependence between the components of a multivariate Markov chain in the context of weak Markovian consistency and strong Markovian consistency. In this connection, we also introduce and discuss the concept of weak Markov copulae.
Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class $\mathcal{C}_{ar}^d$ of all $d$-dimensional Archimedean copulas with $d \geq 3$ and proves the afore-mentioned implication with respect to conditioning on the first $d-1$ coordinates. Several proper\-ties equivalent to pointwise convergence in $\mathcal{C}_{ar}^d$ are established and - as by-product of working with conditional distributions (Markov kernels) - a…
For extreme value copulas with a known upper tail dependence coefficient we find pointwise upper and lower bounds, which are used to establish upper and lower bounds of the Spearman and Kendall correlation coefficients. We shown that in all cases the lower bounds are attained on Marshall--Olkin copulas, and the upper ones, on copulas with piecewise linear dependence functions.
In this thesis, the tail properties of multivariate Archimedean copulas are investigated using known representation theorems involving L1-norm symmetric distributions and the Williamson d-transform. Several new results on the asymptotic properties of the Williamson d-transform are established and subsequently used to study the tails of Archimedean copulas. This makes it possible to recover many known results regarding their tail behavior in a straightforward and transparent way. In particular, coefficients of tail dependence, extreme value limits and threshold copulas are considered. A central…
Finding upper and lower bounds to integrals with respect to copulas is a quite prominent problem in applied probability. In their 2014 paper, Hofer and Iaco showed how particular two dimensional copulas are related to optimal solutions of the two dimensional assignment problem. Using this, they managed to approximate integrals with respect to two dimensional copulas. In this paper, we will further illuminate this connection, extend it to d-dimensional copulas and therefore generalize the method from Hofer and Iaco to arbitrary dimensions. We also provide convergence statements. As an example, …
We define in a probabilistic way a parametric family of multivariate extreme value distributions. We derive its copula, which is a mixture of several complete dependent copulas and total independent copulas, and the bivariate tail dependence and extremal coefficients. Based on the obtained results for these coefficients, we propose a method to built multivariate extreme value distributions with prescribed tail/extremal coefficients. We illustrate the results with examples of simulation of these distributions.
Given two multivariate copulas with corresponding tail dependence functions, we investigate the relation between a natural tail dependence ordering $\leq_{tdo}$ and the order $\leq_{loc}$ of local stochastic dominance. We show that, although the two orderings are not equivalent in general, they coincide for various important classes of copulas, among them all multivariate Archimedean and bivariate lower extreme value copulas. We illustrate the relevance of our results by an implication to risk management.
We show that the set of $d$-variate symmetric stable tail dependence functions, uniquely associated with exchangeable $d$-dimensional extreme-value copulas, is a simplex and 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 $k \geq 1$. 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 conditionall…
We explore the consequences of a set of axioms which extend Scarsini's axioms for bivariate measures of concordance to the multivariate case and exhibit the following results: (1) A method of extending measures of concordance from the bivariate case to arbitrarily high dimensions. (2) A formula expressing the measure of concordance of the random vectors $(\pm X_1,...,\pm X_n)$ in terms of the measures of concordance of the "marginal" random vectors $(X_{i_1},...,X_{i_k})$. (3) A method of expressing the measure of concordance of an odd-dimensional copula in terms of the measures of concordance…
The energy distance and energy scores became important tools in multivariate statistics and multivariate probabilistic forecasting in recent years. They are both based on the expected distance of two independent samples. In this paper we study dependence uncertainty bounds for these quantities under the assumption that we know the marginals but do not know the dependence structure. We find some interesting sharp analytic bounds, where one of them is obtained for an unusual spherically symmetric copula. These results should help to better understand the sensitivity of these measures to misspeci…
A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and lower tails of the copula are classified into one of three classes each, one corresponding to asymptotic dependence and the other two to asymptotic independence. For a long list of single-parameter families, the relevant tail quantities are computed so …
Archimedean copulas are popular in the world of multivariate modelling as a result of their breadth, tractability, and flexibility. A. J. McNeil and J. Nešlehová (2009) showed that the class of Archimedean copulas coincides with the class of multivariate $\ell_1$-norm symmetric distributions. Building upon their results, we introduce a class of multivariate Markov processes that we call `Archimedean survival processes' (ASPs). An ASP is defined over a finite time interval, is equivalent in law to a multivariate gamma process, and its terminal value has an Archimedean survival copula. There exi…
In this paper our aim is to characterize the set of extreme points of the set of all n-dimensional copulas (n > 1). We have shown that a copula must induce a singular measure with respect to Lebesgue measure in order to be an extreme point in the set of n-dimensional copulas. We also have discovered some sufficient conditions for a copula to be an extreme copula. We have presented a construction of a small subset of n-dimensional extreme copulas such that any n-dimensional copula is a limit point of that subset with respect to weak convergence. The applications of such a theory are widespread,…
It is well known and readily seen that the maximum of $n$ independent and uniformly on $[0,1]$ distributed random variables, suitably standardised, converges in total variation distance, as $n$ increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalized Pareto copula. Sklar's theorem then implies convergence in variational distance of the maximum of $n$ independent and identically distributed ran…
