Multivariate Archimedean Copulas Research Articles

On convergence and singularity of conditional copulas of multivariate Archimedean copulas, and conditional dependence

The present paper derives an explicit expression for (a version of) every uni- and multivariate conditional distribution (i.e., Markov kernel) of Archimedean copulas and uses this representation to generalize a recently established result, saying that in the class of multivariate Archimedean copulas standard uniform convergence implies weak convergence of almost all univariate Markov kernels, to arbitrary multivariate Markov kernels. Moreover, it is proved that an Archimedean copula is singular if, and only if, almost all uni- and multivariate Markov kernels are singular. These results are then applied to conditional Archimedean copulas which are reintroduced largely from a Markov kernel perspective and it is shown that convergence, singularity and conditional increasingness carry over from Archimedean copulas to their conditional copulas. As a consequence, the surprising fact is established that estimating (the generator of) an Archimedean copula directly yields an estimator of (the generator of) its conditional copula. Building upon that, we sketch the use and estimation of a conditional version of a recently introduced dependence measure as alternative to well-known conditional versions of association measures in order to study the dependence behavior of Archimedean models when fixing covariate values.

On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform

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 Card of all d-dimensional Archimedean copulas with d≥3 and proves the afore-mentioned implication with respect to conditioning on the first d−1 coordinates. Several properties equivalent to pointwise convergence in Card are established and - as by-product of working with conditional distributions (Markov kernels) - alternative simple proofs for the well-known formulas for the level set masses μC(Lt) and the Kendall distribution function FKd as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators ψ of d-dimensional Archimedean copulas from the perspective of their so-called Williamson measures γ on (0,∞) is then shown to allow not only to derive surprisingly simple expressions for μC(Lt) and FKd in terms of γ and to characterize pointwise convergence in Card by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of γ directly carry over to the corresponding copula Cγ∈Card. These results are finally used to prove the fact that the family of all absolutely continuous and the family of all singular d-dimensional copulas is dense in Card and to underline that despite of their simple algebraic structure Archimedean copulas may exhibit surprisingly singular behavior in the sense of irregularity of their conditional distribution functions.

In-home and out-of-home energy consumption by elderly and non-elderly households: a comparative analysis based on a copula model

This paper compares the energy consumption of elderly and non-elderly households, recognizing the ever-increasing importance of the aging population for energy policy. Decisions of the domestic (in-home) and transport (out-of-home) sectors at the household level are jointly incorporated by building a multivariate Hierarchical Archimedean Copula model with intra-sector and inter-sector dependencies. Data from a life-oriented behavioral survey in Japan are used, which includes 532 elderly households and 340 non-elderly households. A variety of life-related factors, together with built environment factors, are introduced into the above model. Modeling analysis results show that inter- and intra-sector dependencies are heterogeneous across households in terms of these factors. It is found that almost equal numbers of influential factors are sensitive and insensitive to age-based sample segmentation. Various similarities and dissimilarities across energy types between elderly and non-elderly households are revealed. The paper concludes with a discussion of the policy implications of the results.

Using hierarchical Archimedean copulas for modelling mortality dependence and pricing mortality-linked securities

AbstractIn this article, we explore the use of multivariate Archimedean copulas in modelling the mortality dependence between different countries and pricing mortality bonds. We study the fitting performance of multi-dimensional, fully nested, and partially nested Archimedean copulas and test 11 types of generators and two skewed distributions. To evaluate their practical usefulness, we adopt the fitted models to compute the market prices for some typical mortality bond structures. The results show that the copula assumption has a significant impact on the calculation of the prices of mortality-linked securities and the management of extreme mortality risks.

Convergence of Linear Approximation of Archimedean Generator from Williamson’s Transform in Examples

Abstract We discuss a new construction method for obtaining additive generators of Archimedean copulas proposed by McNeil, A. J.-Nešlehová, J.: Multivariate Archimedean copulas, d-monotone functions and l1-norm symmetric distributions, Ann. Statist. 37 (2009), 3059-3097, the so-called Williamson n-transform, and illustrate it by several examples. We show that due to the equivalence of convergences of positive distance functions, additive generators and copulas, we may approximate any n-dimensional Archimedean copula by an Archimedean copula generated by a transformation of weighted sum of Dirac functions concentrated in certain suitable points. Specifically, in two dimensional case this means that any Archimedean copula can be approximated by a piece-wise linear Archimedean copula, moreover the approximation of generator by linear splines circumvents the problem with the non-existence of explicit inverse.

A note on upper-patched generators for Archimedean copulas

The class of multivariate Archimedean copulas is defined by using a real-valued function called the generator of the copula. This generator satisfies some properties, including d-monotonicity. We propose here a new basic transformation of this generator, preserving these properties, thus ensuring the validity of the transformed generator and inducing a proper valid copula. This transformation acts only on a specific portion of the generator, it allows both the non-reduction of the likelihood on a given dataset, and the choice of the upper tail dependence coefficient of the transformed copula. Numerical illustrations show the utility of this construction, which can improve the fit of a given copula both on its central part and its tail.

Application of a nested trivariate copula structure in a competing duration hazard-based vehicle transaction decision model

ABSTRACTVehicle ownership has been the subject of numerous transportation studies, which typically have adopted a variety of econometric frameworks at both aggregate and disaggregate choice modeling levels. These models have covered various aspects of vehicle ownership, including vehicle transaction behavior, number of vehicles held, vehicle type choice, and vehicle use. Vehicle transaction models have been developed in both static and dynamic paradigms. However, traditionally transaction timing has been overlooked in static vehicle ownership models despite the significant role that it plays in explaining vehicle ownership behavior. The decision about transaction type is typically assumed to be independently made; therefore, independent timing models are developed for each transaction type. This study aims to introduce a joint transaction type and timing model at the disaggregate level of individuals using a discrete competing proportional hazard model. The inter-correlation among multiple transaction types is modeled in this study by utilizing a set of multivariate Archimedean copula density functions to approximate the multivariate probability density function for the error terms of transaction types in a fully nested structure.

Distorted Copula-Based Probability Distribution of a Counting Hierarchical Variable: A Credit Risk Application

In this paper, we propose a novel approach for the computation of the probability distribution of a counting variable linked to a multivariate hierarchical Archimedean copula function. The hierarchy has a twofold impact: it acts on the aggregation step but also it determines the arrival policy of the random event. The novelty of this work is to introduce this policy, formalized as an arrival matrix, i.e., a random matrix of dependent 0–1 random variables, into the model. This arrival matrix represents the set of distorted (by the policy itself) combinatorial distributions of the event, i.e., of the most probable scenarios. To this distorted version of the [Formula: see text] approach [see Ref. 7 and Ref. 27], we are now able to apply a pure hierarchical Archimedean dependence structure among variables. As an empirical application, we study the problem of evaluating the probability distribution of losses related to the default of various type of counterparts in a structured portfolio exposed to the credit risk of a selected set among the major banks of European area and to the correlations among these risks.

A directory of families of infinitely extendible Archimedean copulas

This article introduces families of strict multivariate Archimedean copulas, C, the ordering of copulas within these families, and associated boundary copulas. The new families are compared with the Clayton, Frank, and Gumbel families with regard to measures of dependence, including tail dependence, Kendall's tau, and Spearman's rho for dimensions 2, 3, and 4. A new inequality that gives maximal values of C(u,v)−uv is conjectured.

On tail dependence coefficients of transformed multivariate Archimedean copulas

This paper presents the impact of a class of transformations of copulas in their upper and lower multivariate tail dependence coefficients. In particular we focus on multivariate Archimedean copulas. In the first part of this paper, we calculate multivariate transformed tail dependence coefficients when the generator of the considered transformed copula exhibits some regular variation properties, and we investigate the behaviour of these coefficients in cases that are close to tail independence. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. These results are also valid for non-transformed Archimedean copulas. In the second part we deal with transformations presented by Di Bernardino and Rullière [20]. We show the utility of using transformed Archimedean copulas, as they permit to build Archimedean generators exhibiting any chosen couple of lower and upper tail dependence coefficients. Finally, we detail the extreme behaviour of the transformed radial part of Archimedean copulas (using results in Larsson and Nešlehová [52]) and we explain possible applications with transformed Markov chains.

Bayesian Inference of Pair-Copula Constriction for Multivariate Dependency Modeling of Iran’s Macroeconomic Variables

Shahid Chamran University, Ahvaz, Iran Bayesian inference of pair-copula constriction (PCC) is used for multivariate dependency modeling of Iran’s macroeconomics variables: oil revenue, economic growth, total consumption and investment. These constructions are based on bivariate t-copulas as building blocks and can model the nature of extreme events in bivariate margins individually. The model parameter was estimated based on Markov chain Monte Carlo (MCMC) methods. A MCMC algorithm reveals unconditional as well as conditional independence in Iran’s macroeconomic variables, which can simplify resulting PCC’s for these data. Key words: Monte Carlo Markov Chain Method, pair-copula construction, vine. Introduction Multivariate data usually exhibit a complex pattern of dependency. Methods such as graphical model and Bayesian networks are available to investigate dependency structures in multivariate data. One increasingly popular approach for constructing high dimensional dependency is based on copulas. Copulas are multivariate distribution functions with uniform margins which allow representation of joint distribution functions as a function of marginal distributions and a copula (Sklar, 1959). Copulas are used in various fields of applied sciences, but are most widely used in economics, finance and risk management (Embrechts, et al., 2003; Patton, 2004; Nolte, 2008). The class of copulas for bivariate data is rich in comparison to the one for �−dimensional data with � ≥ 3. Untilrecently, Gaussian and t-copulas or, more M. R. Zadkarami is a associated professor on the Faculty of Mathematics and Computer Sciences in the Statistics Department. Email him at: zadkarami@yahoo.co.uk. O. Chatrabgoun is a PhD student on the Faculty of Mathematics and Computer Sciences in the Statistics Department. Email him at: o-chatrabgoun@phdstu.scu.ac.ir. generally, elliptical copulas, have been used for multivariate data (Frahm, et al., 2003). The generalization of bivariate copulas to multivariate copulas of dimensions larger than 2 is not straightforward, however there is one simple generalization for Archimedean copulas known as exchangeable Archimedean copulas (Frey & McNeil, 2003). It should be noted that not all bivariate Archimedean copulas have a corresponding multivariate exchangeable version (Nelsen, 1999). Approaches for constructing multivariate Archimedean copulas of more than 2 have dimensions been developed by Joe (1997), Embrechts, et al. (2003), Whelan, (2004), McNeil, et al. (2006), Savu and Trede (2006) and McNeil (2007). Joe (1996) and Bedford and Cooke (2001, 2002) constructed flexible higher-dimensional copulas by using only bivariate copulas as building blocks, which they termed vines. Kurowicka and Cooke (2006) discussed Gaussian vine constructions in details. Aas, et al. (2007) first recognized the general construction principle for deriving multivariate copulas; they used more general bivariate copulas than the Gaussian copula and applied these construction methods to financial risk data using more appropriate pair-copulas such as the bivariate t Clayton and Gumbel copulas. According to recent empirical investigations of Berg and Aas (2007) and Fischer, et al. (2007), the vine constructions based on bivariate t-

Archimedean copula-based estimation of distribution algorithm for multi-objective optimisation

Estimation of distribution algorithms (EDAs) are a class of evolutionary optimisation algorithms based on probability distribution model. This article extends the basic EDAs for tackling multi-objective optimisation problems by incorporating Archimedean copulas for constructing probability distribution model, and using the concept of Pareto dominance. In the algorithm, the marginal distributions from the current population are firstly estimated by kernel estimation method and are used to estimate the parameter of the Archimedean copula function generator by using the maximum likelihood method. Afterwards, the multivariate Archimedean copula sample algorithm is used to generate current offsprings population by sampling the n-dimensional Laplace transform Archimedean copula. The population with the current population and current offsprings population is sorted based on non-domination, and the best individuals are selected to form the next population based on rank and the crowding distance. The proposed algorithm is tested to compare with NSGA-II, PAES and SPEA2 using a set of benchmark functions. Both convergence and diversity metrics are used to evaluate the performance of the algorithm. The experimental results show that the algorithm outperforms NSGA-II, PAES and SPEA2 in two metrics.

Inference in multivariate Archimedean copula models

This paper proposes new rank-based estimators for multivariate Archimedean copulas. The approach stems from a recent representation of these copulas as the survival copulas of simplex distributions. The procedures are based on a reconstruction of the radial part of the simplex distribution from the Kendall distribution, which arises through the multivariate probability integral transformation of the data. In the bivariate case, the methodology is justified by the well known fact that an Archimedean copula is in one-to-one correspondence with its Kendall distribution. It is proved here that this property continues to hold in the trivariate case, and strong evidence is provided that it extends to any dimension. In addition, a criterion is derived for the convergence of sequences of multivariate Archimedean copulas. This result is then used to show consistency of the proposed estimators.

Tail order and intermediate tail dependence of multivariate copulas

In order to study copula families that have tail patterns and tail asymmetry different from multivariate Gaussian and t copulas, we introduce the concepts of tail order and tail order functions. These provide an integrated way to study both tail dependence and intermediate tail dependence. Some fundamental properties of tail order and tail order functions are obtained. For the multivariate Archimedean copula, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed from the Laplace transform of the random variable, and extend the results of Charpentier and Segers [7] [A. Charpentier, J. Segers, Tails of multivariate Archimedean copulas, Journal of Multivariate Analysis 100 (7) (2009) 1521–1537] for upper tails of Archimedean copulas. In addition, a new one-parameter Archimedean copula family based on the Laplace transform of the inverse Gamma distribution is proposed; it possesses patterns of upper and lower tails not seen in commonly used copula families. Finally, tail orders are studied for copulas constructed from mixtures of max-infinitely divisible copulas.

Efficient algorithms for basket default swap pricing with multivariate Archimedean copulas

We introduce a new importance sampling method for pricing basket default swaps employing exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structures than existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of the proposed algorithms, we present numerical examples and compare them with the crude Monte Carlo simulation, and finally show that our proposed estimators produce considerably smaller variances.

Multivariate Archimedean copulas, d-monotone functions and ℓ1-norm symmetric distributions

It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a d-dimensional copula is that the generator is a d-monotone function. The class of d-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of d-dimensional l 1 ; -norm symmetric distributions that place no point mass at the origin. The d-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189-207] in an analogous manner to the well-known Bernstein-Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the d-dimensional Kendall function and Kendall's rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.

Efficient Algorithms for Basket Default Swap Pricing with Multivariate Archimedean Copulas

We introduce a new importance sampling method for pricing basket default swaps based on exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structure than the existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of our proposed algorithms, we demonstrate several numerical examples compared with the crude Monte Carlo simulation, and show that our proposed estimators produce remarkably small variance with accurately expected values in pricing basket default swaps.

Tools for Sampling Multivariate Archimedean Copulas

A hurdle for practical implementation of any multivariate Archimedean copula was the absence of an efficient method for generating them. The most frequently used approach named conditional distribution one, involves differentiation step for each dimension of the problem. For this reason, it is not feasible in higher dimension. Marshall and Olkin proposed an alternative method, which is computationally more straightforward than the conditional distribution approach. We present the tools necessary for understand it and use it. We introduce the Laplace Transform and its role in the generation of multivariate Archimedean copulas. In order to cover the gap between the theory and its practical implementation VBA code and R one are provided.

GeD spline estimation of multivariate Archimedean copulas

A new multivariate Archimedean copula estimation method is proposed in a non-parametric setting. The method uses the so-called Geometrically Designed splines (GeD splines) to represent the cdf of a random variable W θ , obtained through the probability integral transform of an Archimedean copula with parameter θ . Sufficient conditions for the GeD spline estimator to possess the properties of the underlying theoretical cdf, K ( θ , t ) , of W θ , are given. The latter conditions allow for defining a three-step estimation procedure for solving the resulting non-linear regression problem with linear inequality constraints. In the proposed procedure, finding the number and location of the knots and the coefficients of the unconstrained GeD spline estimator and solving the constraint least-squares optimisation problem are separated. Thus, the resulting spline estimator K ˆ ( θ ˆ , t ) is used to recover the generator and the related Archimedean copula by solving an ordinary differential equation. The proposed method is truly multivariate, it brings about numerical efficiency and as a result can be applied with large volumes of data and for dimensions d ≥ 2 , as illustrated by the numerical examples presented.

Simulating from Exchangeable Archimedean Copulas

Multivariate exchangeable Archimedean copulas are one of the most popular classes of copulas that are used in actuarial science and finance for modeling risk dependencies and for using them to quantify the magnitude of tail dependence. Owing to the increase in popularity of copulas to measure dependent risks, generating from multivariate copulas has become a very crucial exercise. Current methods for generating from multivariate Archimedean copulas could become a very difficult task as the number of dimension increases. The resulting analytical procedures suggested in the existing literature do not offer much guidance for practical implementation. This article presents an algorithm for generating values from the multivariate exchangeable Archimedean copulas based on a multivariate extension of a bivariate result. A procedure for generating values from bivariate Archimedean copulas has been originally proposed in Genest and Rivest (1993) and again later described in Nelsen (1999) and Embrechts et al. (2001ab). Using a proof that is simply based on fundamental Jacobian techniques for deriving distributions of transformed random variables, we are able to extend the bivariate result into the multivariate case allowing us to develop an interesting algorithm to generate values from exchangeable Archimedean copulas. As auxiliary results, we are able to derive the distribution function of an n-dimensional Archimedean copula, a result already known in Genest and Rivest (2001), but our approach of proving this result is based on a different perspective. This article focuses on this class of copulas that has one generating function and one parameter that characterizes the dependence structure of the joint distribution function.