Bivariate Margins Research Articles

On a Family of Log-Gamma-Generated Archimedean Copulas

Modeling dependence structure among various risks, especially the measure of tail dependence and the aggregation of risks, is crucial for risk management. In this article, we present an extension to the traditional one-parameter Archimedean copulas by integrating the log-gamma-generated (LGG) margins. This class of novel multivariate distribution can better capture the tail dependence. The distortion effect on the classic one-parameter Archimedean copulas is well exhibited and the analytical expression of the sum of bivariate margins is proposed. The model provides a flexible way to capture tail risks and aggregate portfolio losses. Sufficient conditions for constructing a legitimate d-dimensional LGG Archimedean copula as well as the simulation framework are also proposed. Furthermore, two applications of this model are presented using concrete insurance datasets.

On a construction of multivariate distributions given some multidimensional marginals

AbstractIn this paper we investigate the link between the joint law of a d-dimensional random vector and the law of some of its multivariate marginals. We introduce and focus on a class of distributions, that we call projective, for which we give detailed properties. This allows us to obtain necessary conditions for a given construction to be projective. We illustrate our results by proposing some theoretical projective distributions, as elliptical distributions or a new class of distribution having given bivariate margins. In the case where the data does not necessarily correspond to a projective distribution, we also explain how to build proper distributions while checking that the distance to the prescribed projections is small enough.

A simple non-parametric goodness-of-fit test for elliptical copulas

Abstract In this paper, we propose a simple non-parametric goodness-of-fit test for elliptical copulas of any dimension. It is based on the equality of Kendall’s tau and Blomqvist’s beta for all bivariate margins. Nominal level and power of the proposed test are investigated in a Monte Carlo study. An empirical application illustrates our goodness-of-fit test at work.

Examination and visualisation of the simplifying assumption for vine copulas in three dimensions

SummaryVine copulas are a highly flexible class of dependence models, which are based on the decomposition of the density into bivariate building blocks. For applications one usually makes the simplifying assumption that copulas of conditional distributions are independent of the variables on which they are conditioned. However this assumption has been criticised for being too restrictive. We examine both simplified and non‐simplified vine copulas in three dimensions and investigate conceptual differences. We show and compare contour surfaces of three‐dimensional vine copula models, which prove to be much more informative than the contour lines of the bivariate marginals. Our investigation shows that non‐simplified vine copulas can exhibit arbitrarily irregular shapes, whereas simplified vine copulas appear to be smooth extrapolations of their bivariate margins to three dimensions. In addition to a variety of constructed examples, we also investigate a three‐dimensional subset of the well‐known uranium data set and visually detect the fact that a non‐simplified vine copula is necessary to capture its complex dependence structure.

Samurai Sudoku-Based Space-Filling Designs for Data Pooling

Pooling data from multiple sources plays an increasingly vital role in today’s world. By using a popular Sudoku game, we propose a new type of design, called a Samurai Sudoku-based space-filling design to address this issue. Such a design is an orthogonal array-based Latin hypercube design with the following attractive properties: (i) the complete design achieves uniformity in both univariate and bivariate margins; (ii) it can be divided into groups of subdesigns with overlaps such that each subdesign achieves uniformity in both univariate and bivariate margins; and (iii) each of the overlaps achieves uniformity in both univariate and bivariate margins. Examples are given to illustrate the properties of the proposed design, and to demonstrate the advantages of using the proposed design for pooling data from multiple sources.[Received August 2013. Revised July 2015.]

Spatial Tail Dependence and Survival Stability in a Class of Archimedean Copulas

This paper investigates properties of extensions of tail dependence of Archimax copulas to high dimensional analysis in a spatialized framework. Specifically, we propose a characterization of bivariate margins of spatial Archimax processes while spatial multivariate upper and lower tail dependence coefficients are modeled, respectively, for Archimedean copulas and Archimax ones. A property of stability is given using convex transformations of survival copulas in a spatialized Archimedean family.

AnM-Estimator of Spatial Tail Dependence

SummaryTail dependence models for distributions attracted to a max-stable law are fitted by using observations above a high threshold. To cope with spatial, high dimensional data, a rank-based M-estimator is proposed relying on bivariate margins only. A data-driven weight matrix is used to minimize the asymptotic variance. Empirical process arguments show that the estimator is consistent and asymptotically normal. Its finite sample performance is assessed in simulation experiments involving popular max-stable processes perturbed with additive noise. An analysis of wind speed data from the Netherlands illustrates the method.

Samurai Sudoku-Based Space-Filling Designs for Multi-Source Inference

Pooling data from multiple sources plays an increasingly vital role in today's world. We propose a new type of design, called a Samurai Sudoku-based space-filling design to address this issue. Such a design is an orthogonal array based Latin hypercube design with the following attractive properties: (1) the complete design achieves attractive uniformity in both univariate and bivariate margins; (2) it can be divided into groups of subdesigns with overlaps such that each subdesign achieves maximum uniformity in both univariate and bivariate margins; (3) each of the overlaps achieves maximum uniformity in both univariate and bivariate margins. Examples are given to illustrate the properties of the proposed design, and to demonstrate the advantages of using the proposed design for pooling data from multiple sources.

An M-Estimator of Spatial Tail Dependence

Tail dependence models for distributions attracted to a max-stable law are fitted using observations above a high threshold. To cope with spatial, high-dimensional data, a rank based M-estimator is proposed relying on bivariate margins only. A data-driven weight matrix is used to minimize the asymptotic variance. Empirical process arguments show that the estimator is consistent and asymptotically normal. Its finite-sample performance is assessed in simulation experiments involving popular max-stable processes perturbed with additive noise. An analysis of wind speed data from the Netherlands illustrates the method.

Nonparametric estimation of the tree structure of a nested Archimedean copula

One of the features inherent in nested Archimedean copulas, also called hierarchical Archimedean copulas, is their rooted tree structure. A nonparametric, rank-based method to estimate this structure is presented. The idea is to represent the target structure as a set of trivariate structures, each of which can be estimated individually with ease. Indeed, for any three variables there are only four possible rooted tree structures and, based on a sample, a choice can be made by performing comparisons between the three bivariate margins of the empirical distribution of the three variables. The set of estimated trivariate structures can then be used to build an estimate of the target structure. The advantage of this estimation method is that it does not require any parametric assumptions concerning the generator functions at the nodes of the tree.

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-

Properties of hierarchical Archimedean copulas

Abstract In this paper we analyse the properties of hierarchical Archimedean copulas. This class is a generalisation of the Archimedean copulas and allows for general non-exchangeable dependency structures. We show that the structure of the copula can be uniquely recovered from all bivariate margins. We derive the distribution of the copula values, which is particularly useful for tests and constructing confidence intervals. Furthermore, we analyse dependence orderings, multivariate dependence measures, and extreme value copulas. We pay special attention to the tail dependencies and derive several tail dependence indices for general hierarchical Archimedean copulas.

Limited-information goodness-of-fit testing of hierarchical item factor models.

In applications of item response theory, assessment of model fit is a critical issue. Recently, limited-information goodness-of-fit testing has received increased attention in the psychometrics literature. In contrast to full-information test statistics such as Pearson's X(2) or the likelihood ratio G(2) , these limited-information tests utilize lower-order marginal tables rather than the full contingency table. A notable example is Maydeu-Olivares and colleagues'M2 family of statistics based on univariate and bivariate margins. When the contingency table is sparse, tests based on M2 retain better Type I error rate control than the full-information tests and can be more powerful. While in principle the M2 statistic can be extended to test hierarchical multidimensional item factor models (e.g., bifactor and testlet models), the computation is non-trivial. To obtain M2 , a researcher often has to obtain (many thousands of) marginal probabilities, derivatives, and weights. Each of these must be approximated with high-dimensional numerical integration. We propose a dimension reduction method that can take advantage of the hierarchical factor structure so that the integrals can be approximated far more efficiently. We also propose a new test statistic that can be substantially better calibrated and more powerful than the original M2 statistic when the test is long and the items are polytomous. We use simulations to demonstrate the performance of our new methods and illustrate their effectiveness with applications to real data.

Sudoku-based space-filling designs

Sudoku is played by millions of people across the globe. It has simple rules and is very addictive. The game board is a nine-by-nine grid of numbers from one to nine. Several entries within the grid are provided and the remaining entries must be filled in subject to no row, column, or three-by-three subsquare containing duplicate numbers. By exploiting these three types of uniformity, we propose an approach to constructing a new type of design, called a Sudoku-based space-filling design. Such a design can be divided into groups of subdesigns so that the complete design and each subdesign achieve maximum uniformity in univariate and bivariate margins. Examples are given illustrating the proposed construction method. Applications of such designs include computer experiments with qualitative and quantitative factors, linking parameters in engineering and crossvalidation. Copyright 2011, Oxford University Press.

Nested orthogonal array-based Latin hypercube designs

We propose two methods for constructing a new type of design, called a nested orthogonal array-based Latin hypercube design, intended for multi-fidelity computer experiments. Such designs are two nested space-filling designs in which the large design achieves stratification in both bivariate and univariate margins and the small design achieves stratification in univariate margins. These designs have better space-filling properties than nested Latin hypercube designs in which the large design possesses uniformity in univariate margins only. The first method expands an ordinary Latin hypercube design to a larger design that achieves uniformity in any one- or two-dimensional projection. The second method uses an orthogonal array with strength two to simultaneously construct a pair of nested orthogonal array-based Latin hypercube designs. Examples are given to illustrate the proposed methods. Sampling properties of the proposed designs are derived. Copyright 2011, Oxford University Press.

On weighting of bivariate margins in pairwise likelihood

Composite and pairwise likelihood methods have recently been increasingly used. For clustered data with varying cluster sizes, we study asymptotic relative efficiencies for various weighted pairwise likelihoods, with weight being a function of cluster size. For longitudinal data, we also study weighted pairwise likelihoods with weights that can depend on lag. Good choice of weights are needed to avoid the undesirable behavior of estimators with low efficiency. Some analytic results are obtained using the multivariate normal distribution. For clustered data, a practically good choice of weight is obtained after study of relative efficiencies for an exchangeable multivariate normal model; they are different from weights that had previously been suggested. For longitudinal data, there are advantages to only include bivariate margins of adjacent or nearly adjacent pairs in the weighted pairwise likelihood.

Multivariate Fréchet copulas and conditional value‐at‐risk

Based on the method of copulas, we construct a parametric family of multivariate distributions using mixtures of independent conditional distributions. The new family of multivariate copulas is a convex combination of products of independent and comonotone subcopulas. It fulfills the four most desirable properties that a multivariate statistical model should satisfy. In particular, the bivariate margins belong to a simple but flexible one‐parameter family of bivariate copulas, called linear Spearman copula, which is similar but not identical to the convex family of Fréchet. It is shown that the distribution and stop‐loss transform of dependent sums from this multivariate family can be evaluated using explicit integral formulas, and that these dependent sums are bounded in convex order between the corresponding independent and comonotone sums. The model is applied to the evaluation of the economic risk capital for a portfolio of risks using conditional value‐at‐risk measures. A multivariate conditional value‐at‐risk vector measure is considered. Its components coincide for the constructed multivariate copula with the conditional value‐at‐risk measures of the risk components of the portfolio. This yields a “fair” risk allocation in the sense that each risk component becomes allocated to its coherent conditional value‐at‐risk.

Modelling failure-time associations in data with multiple levels of clustering

SUMMARY In recent years substantial research has been devoted to developing failure-time methodology which accounts for possible dependency between observations. An example is the univariate frailty model (Vaupel, Manton & Stallard, 1979), which incorporates an exchangeable dependence structure by the inclusion of cluster-specific random effects. In some studies it may be reasonable to expect more than one level of within-cluster association: for instance, association between a parent and child versus that between two siblings in studies of familial disease aggregation, or association between two village residents who live in different households versus that between residents of the same household in intervention studies. We propose a family of distributional models for failure-time data that accounts for multiple levels of clustering and reduces in the case of a single clustering level to a univariate frailty model. The resulting survival functions are constructed by a recursive nesting of univariate frailty-type distributions through which archimedean copula forms are determined for all bivariate margins. Properties of the proposed model are developed, illustrated and briefly contrasted with multivariate frailty model properties. In conclusion, we outline the application of our model to marginal risk regression problems.

A strategy for constructing multivariate distributions

Given a random vector (X1,…, Xn) for which the univariate and bivariate marginal distributions belong to some specified families of distributions, we present a procedure for constructing families of multivariate distributions with the specified univariate and bivariate margins. Some general properties of the resulting families of multivariate distributions are reviewed. This procedure is illustrated by generalizing the bivariate Plackett (1965) and Clayton (1978) distributions to three dimensions. In addition to providing rich families of models for data analysis, this method of construction provides a convenient way of simulating observations from multivariate distributions with specific types of univariate and bivariate marginal distributions. A general algorithm for simulating random observations from these families of multivariate distributions is presented

The extremal index for a Markov chain

The paper presents a method of computing the extremal index for a discrete-time stationary Markov chain in continuous state space. The method is based on the assumption that bivariate margins of the process are in the domain of attraction of a bivariate extreme value distribution. Scaling properties of bivariate extremes then lead to a random walk representation for the tail behaviour of the process, and hence to computation of the extremal index in terms of the fluctuation properties of that random walk. The result may then be used to determine the asymptotic distribution of extreme values from the Markov chain.