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

Uyttendaele, Nathan
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
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.

## Multivariate Spearman's rho for aggregating ranks using copulas

Bedo, Justin; Ong, Cheng Soon
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.95%
We study the problem of rank aggregation: given a set of ranked lists, we want to form a consensus ranking. Furthermore, we consider the case of extreme lists: i.e., only the rank of the best or worst elements are known. We impute missing ranks by the average value and generalise Spearman's \rho to extreme ranks. Our main contribution is the derivation of a non-parametric estimator for rank aggregation based on multivariate extensions of Spearman's \rho, which measures correlation between a set of ranked lists. Multivariate Spearman's \rho is defined using copulas, and we show that the geometric mean of normalised ranks maximises multivariate correlation. Motivated by this, we propose a weighted geometric mean approach for learning to rank which has a closed form least squares solution. When only the best or worst elements of a ranked list are known, we impute the missing ranks by the average value, allowing us to apply Spearman's \rho. Finally, we demonstrate good performance on the rank aggregation benchmarks MQ2007 and MQ2008.

## A goodness-of-fit test for bivariate extreme-value copulas

Genest, Christian; Kojadinovic, Ivan; Nešlehová, Johanna; Yan, Jun
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
It is often reasonable to assume that the dependence structure of a bivariate continuous distribution belongs to the class of extreme-value copulas. The latter are characterized by their Pickands dependence function. In this paper, a procedure is proposed for testing whether this function belongs to a given parametric family. The test is based on a Cram\'{e}r--von Mises statistic measuring the distance between an estimate of the parametric Pickands dependence function and either one of two nonparametric estimators thereof studied by Genest and Segers [Ann. Statist. 37 (2009) 2990--3022]. As the limiting distribution of the test statistic depends on unknown parameters, it must be estimated via a parametric bootstrap procedure, the validity of which is established. Monte Carlo simulations are used to assess the power of the test and an extension to dependence structures that are left-tail decreasing in both variables is considered.; Comment: Published in at http://dx.doi.org/10.3150/10-BEJ279 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

## Generation of degree-correlated networks using copulas

Raschke, Mathias; Schläpfer, Markus; Trantopoulos, Konstantinos
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.95%
Dynamical processes on complex networks such as information propagation, innovation diffusion, cascading failures or epidemic spreading are highly affected by their underlying topologies as characterized by, for instance, degree-degree correlations. Here, we introduce the concept of copulas in order to artificially generate random networks with an arbitrary degree distribution and a rich a priori degree-degree correlation (or `association') structure. The accuracy of the proposed formalism and corresponding algorithm is numerically confirmed. The derived network ensembles can be systematically deployed as proper null models, in order to unfold the complex interplay between the topology of real networks and the dynamics on top of them.

## Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas

Wegkamp, Marten; Zhao, Yue
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.95%
We study the adaptive estimation of copula correlation matrix $\Sigma$ for elliptical copulas. In this context, the correlations are connected to Kendall's tau through a sine function transformation. Hence, a natural estimate for $\Sigma$ is the plug-in estimator $\widehat\Sigma$ with Kendall's tau statistic. We first obtain a sharp bound for the operator norm of $\widehat \Sigma - \Sigma$. Then, we study a factor model for $\Sigma$, for which we propose a refined estimator $\widetilde\Sigma$ by fitting a low-rank matrix plus a diagonal matrix to $\widehat\Sigma$ using least squares with a nuclear norm penalty on the low-rank matrix. The bound for the operator norm of $\widehat \Sigma - \Sigma$ serves to scale the penalty term, and we obtain finite sample oracle inequalities for $\widetilde\Sigma$. We also consider an elementary factor model of $\Sigma$, for which we propose closed-form estimators. We provide data-driven versions for all our estimation procedures and performance bounds.

## Estimators for Archimedean copulas in high dimensions

Hofert, Marius; Maechler, Martin; McNeil, Alexander J.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.95%
The performance of known and new parametric estimators for Archimedean copulas is investigated, with special focus on large dimensions and numerical difficulties. In particular, method-of-moments-like estimators based on pairwise Kendall's tau, a multivariate extension of Blomqvist's beta, minimum distance estimators, the maximum-likelihood estimator, a simulated maximum-likelihood estimator, and a maximum-likelihood estimator based on the copula diagonal are studied. Their performance is compared in a large-scale simulation study both under known and unknown margins (pseudo-observations), in small and high dimensions, under small and large dependencies, various different Archimedean families and sample sizes. High dimensions up to one hundred are considered for the first time and computational problems arising from such large dimensions are addressed in detail. All methods are implemented in the open source \R{} package \pkg{copula} and can thus be easily accessed and studied.

## A new bivariate extension of FGM copulas

Amblard, Cécile; Girard, Stéphane
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
We propose a new family of copulas generalizing the Farlie-Gumbel-Morgenstern family and generated by two univariate functions. The main feature of this family is to permit the modeling of high positive dependence. In particular, it is established that the range of the Spearman's Rho is [-3/4,1] and that the upper tail dependence coefficient can reach any value in [0,1]. Necessary and sufficient conditions are given on the generating functions in order to obtain various dependence properties. Some examples of parametric subfamilies are provided.

## Model-based clustering of Gaussian copulas for mixed data

Marbac, Matthieu; Biernacki, Christophe; Vandewalle, Vincent
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.95%
Clustering task of mixed data is a challenging problem. In a probabilistic framework, the main difficulty is due to a shortage of conventional distributions for such data. In this paper, we propose to achieve the mixed data clustering with a Gaussian copula mixture model, since copulas, and in particular the Gaussian ones, are powerful tools for easily modelling the distribution of multivariate variables. Indeed, considering a mixing of continuous, integer and ordinal variables (thus all having a cumulative distribution function), this copula mixture model defines intra-component dependencies similar to a Gaussian mixture, so with classical correlation meaning. Simultaneously, it preserves standard margins associated to continuous, integer and ordered features, namely the Gaussian, the Poisson and the ordered multinomial distributions. As an interesting by-product, the proposed mixture model generalizes many well-known ones and also provides tools of visualization based on the parameters. At a practical level, the Bayesian inference is retained and it is achieved with a Metropolis-within-Gibbs sampler. Experiments on simulated and real data sets finally illustrate the expected advantages of the proposed model for mixed data: flexible and meaningful parametrization combined with visualization features.

## Shaping tail dependencies by nesting box copulas

Hummel, Christoph
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.95%
We introduce a family of copulas which are locally piecewise uniform in the interior of the unit cube of any given dimension. Within that family, the simultaneous control of tail dependencies of all projections to faces of the cube is possible and we give an efficient sampling algorithm. The combination of these two properties may be appealing to risk modellers.; Comment: 25 pages, 3 figures, added reference, remarks in Section 6, corrected typos

## Additive Models for Conditional Copulas

Sabeti, Avideh; Wei, Mian; Craiu, Radu V.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
Conditional copulas are flexible statistical tools that couple joint conditional and marginal conditional distributions. In a linear regression setting with more than one covariate and two dependent outcomes, we propose the use of additive models for conditional bivariate copula models and discuss computation and model selection tools for performing Bayesian inference. The method is illustrated using simulations and a real example.

## Time and Space Varying Copulas

Crane, Glenis
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
In this article we review existing literature on dynamic copulas and then propose an n-copula which varies in time and space. Our approach makes use of stochastic differential equations, and gives rise to a dynamic copula which is able to capture the dependence between multiple Markov diffusion processes. This model is suitable for pricing basket derivatives in finance and may also be applicable to other areas such as bioinformatics and environmental science.

## Computation of copulas by Fourier methods

Papapantoleon, Antonis
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.95%
We provide an integral representation for the (implied) copulas of dependent random variables in terms of their moment generating functions. The proof uses ideas from Fourier methods for option pricing. This representation can be used for a large class of models from mathematical finance, including L\'evy and affine processes. As an application, we compute the implied copula of the NIG L\'evy process which exhibits notable time-dependence.; Comment: 7 pages, 3 figures

## Copulas in three dimensions with prescribed correlations

Devroye, Luc; Letac, Gerard
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
Given an arbitrary three-dimensional correlation matrix, we prove that there exists a three-dimensional joint distribution for the random variable $(X,Y,Z)$ such that $X$,$Y$ and $Z$ are identically distributed with beta distribution $\beta_{k,k}(dx)$ on $(0,1)$ if $k\geq 1/2$. This implies that any correlation structure can be attained for three-dimensional copulas.; Comment: 15 pages, 2 figures

## Using a priori knowledge to construct copulas

Mari, Dominique Drouet; Monbet, Valerie
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
Our purpose is to model the dependence between two random variables, taking into account a priori knowledge on these variables. For example, in many applications (oceanography, finance...), there exists an order relation between the two variables; when one takes high values, the other cannot take low values, but the contrary is possible. The dependence for the high values of the two variables is, therefore, not symmetric. However a minimal dependence also exists: low values of one variable are associated with low values of the other variable. The dependence can also be extreme for the maxima or the minima of the two variables. In this paper, we construct step by step asymmetric copulas with asymptotic minimal dependence, and with or without asymptotic maximal dependence, using mixture variables to get at first asymmetric dependence and then minimal dependence. We fit these models to a real dataset of sea states and compare them using Likelihood Ratio Tests when they are nested, and BIC- criterion (Bayesian Information criterion) otherwise.

## Semi-Supervised Domain Adaptation with Non-Parametric Copulas

Lopez-Paz, David; Hernández-Lobato, José Miguel; Schölkopf, Bernhard
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
A new framework based on the theory of copulas is proposed to address semi- supervised domain adaptation problems. The presented method factorizes any multivariate density into a product of marginal distributions and bivariate cop- ula functions. Therefore, changes in each of these factors can be detected and corrected to adapt a density model accross different learning domains. Impor- tantly, we introduce a novel vine copula model, which allows for this factorization in a non-parametric manner. Experimental results on regression problems with real-world data illustrate the efficacy of the proposed approach when compared to state-of-the-art techniques.; Comment: 9 pages, Appearing on Advances in Neural Information Processing Systems 25

## Modeling covariate-contingent correlation and tail-dependence with copulas

Li, Feng
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
Copulas provide an attractive approach for constructing multivariate densities with flexible marginal distributions and different forms of dependence. Of particular importance in many areas is the possibility of explicitly modeling tail-dependence. Most of the available approaches estimate tail-dependence and correlations via nuisance parameters, yielding results that are neither tractable nor interpretable for practitioners. We propose a general Bayesian approach for directly modeling tail-dependence and correlations as explicit functions of covariates. Our method allows for variable selection among the covariates in the marginal models and in the copula parameters. Posterior inference is carried out using a novel and efficient MCMC simulation method.

## On mixtures of copulas and mixing coefficients

Longla, Martial
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.95%
We show that if the density of the absolutely continuous part of a copula is bounded away from zero on a set of Lebesgue measure 1, then that copula generates \textquotedblleft lower $\psi$-mixing\textquotedblright\ stationary Markov chains. This conclusion implies $\phi$-mixing, $\rho$-mixing, $\beta$-mixing and \textquotedblleft interlaced $\rho$-mixing\textquotedblright . We also provide some new results on the mixing structure of Markov chains generated by mixtures of copulas.; Comment: 11pages

## Gaussian Process Conditional Copulas with Applications to Financial Time Series

Hernández-Lobato, José Miguel; Lloyd, James Robert; Hernández-Lobato, Daniel
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.95%
The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be inaccurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables. We evaluate the ability of our method to predict time-varying dependencies on several equities and currencies and observe consistent performance gains compared to static copula models and other time-varying copula methods.

## When does the stock market listen to economic news? New evidence from copulas and news wires

Medovikov, Ivan
Tipo: Artigo de Revista Científica