Nested Archimedean copulas: a new class of nonparametric tree structure estimators

Any nested Archimedean copula is defined starting from a rooted phylogenetic tree, for which a new class of nonparametric estimators is presented. An estimator from this new class relies on a two-step procedure where first a binary tree is built and second is collapsed if necessary to give an estimate of the target tree structure. Several examples of estimators from this class are given and the performance of each of these estimators, as well as of the only known comparable estimator, is assessed by means of a simulation study involving target structures in various dimensions, showing that the new estimators, besides being faster, usually offer better performance as well. Further, among the given examples of estimators from the new class, one of the best performing one is applied on three datasets: 482 students and their results to various examens, 26 European countries in 1979 and the percentage of workers employed in different economic activities, and 104 countries in 2002 for which various health-related variables are available. The resulting estimated trees offer valuable insights on the analyzed data. The future of nested Archimedean copulas in general is also discussed

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.Comment: 25 pages, 9 figure

On the estimation of nested Archimedean copulas: A theoretical and an experimental comparison

A lot of progress regarding estimation of nested Archimedean cop- ulas has been booked since their introduction by Joe (1997). The currently published procedures can be seen as particular cases of two different, more general, approaches. In the first approach, the tree structure of the target nested Archimedean copulas is estimated us- ing hierarchical clustering to get a binary tree and then parts of this binary tree are collapsed according to some strategy. This two-step estimation of the tree structure paves the way for an easy estimation of the generators afterwards. In contrast to the first approach, the sec- ond approach estimates the tree structure free of any concern for the generators. While this is the main strength of this second approach, it is also its main weakness: estimation of the generators afterwards still lacks a solution. In this paper, both approaches are formally explored, detailed explanations and examples are given, as well as results from a performance study where a new way of comparing tree structure esti- mators is offered. A nested Archimedean copula is also estimated based on exams results from 482 students, and a naive attempt to check the fit is made using principal component analysis

High-dimensional dependence modeling using copulas

Copulas have been introduced more than half a century ago and represent a significant breakthrough in the study of dependencies between random variables, as they allow to do so free of any concern for the univariate margins, which, by definition, have nothing to do with the way the random variables interact with one another. However, while the framework for copulas is well established, the problem of finding actual copulas remains. The development of bivariate copulas (d=2) toke off during the last decades, but satisfying multidimensional copulas (d>2) are still lacking, with the possible exception of vine copulas. In this thesis, the development of multidimensional copulas is pushed further, in an effort to better understand multidimensional random phenoma. This includes new developments for a particular kind of graphs, called phylogenetic trees, research on nested Archimedean copulas, and the extension of factor copulas.(SC - Sciences) -- UCL, 201

Building conditionally dependent parametric one-factor copulas

So far, one-factor copulas induce conditional independence with respect to a latent factor. In this paper, we extend one-factor copulas to conditionally dependent models. This is achieved through two representations which allow to build new parametric one-factor copulas with a varying conditional dependence structure. Moreover, the latent factor's distribution can be estimated despite it being unobserved. In order to dis- tinguish between conditionally independent and conditionally dependent one-factor copulas, we provide with a novel statistical test which does not assume any parametric form for the conditional dependence structure. Illustrations of the approach are provided through examples, numerical experiments as well as a real data analysis where we capture the intrinsic state of a financial market and the dependence structure of its individual assets