Although copulas are used and defined for various infinite-dimensional objects (e.g. Gaussian processes and Markov processes), there is no prevalent notion of a copula that unifies these concepts. We propose a unified approach and define copulas as probability measures on general product spaces. For this we prove Sklar's Theorem in this infinite-dimensional setting. We show how to transfer this result to various function space settings and describe how to model and approximate dependent probability measures in these spaces in the realm of copulas.
Copula models are flexible tools to represent complex structures of dependence for multivariate random variables. According to Sklar's theorem (Sklar, 1959), any d-dimensional absolutely continuous density can be uniquely represented as the product of the marginal distributions and a copula function which captures the dependence structure among the vector components. In real data applications, the interest of the analyses often lies on specific functionals of the dependence, which quantify aspects of it in a few numerical values. A broad literature exists on such functionals, however extension…
Sklar's theorem is an important tool that connects bidimensional distribution functions with their marginals by means of a copula. When there is imprecision about the marginals, we can model the available information by means of p-boxes, that are pairs of ordered distribution functions. Similarly, we can consider a set of copulas instead of a single one. We study the extension of Sklar's theorem under these conditions, and link the obtained results to stochastic ordering with imprecision.
In this article, the concept of copulas is generalised to infinite dimensional Hilbert spaces. We show one direction of Sklar's theorem and explain that the other direction fails in infinite dimensional Hilbert spaces. We derive a necessary and sufficient condition which allows to state this direction of Sklar's theorem in Hilbert spaces. We consider copulas with densities and specifically construct copulas in a Hilbert space by a family of pairwise copulas with densities.
In this note we provide a quick proof of the Sklar's Theorem on the existence of copulas by using the generalized inverse functions as in the one dimensional case, but a little more sophisticated.
We consider multivariate copula-based stationary time-series under Gaussian subordination. Observed time series are subordinated to long-range dependent Gaussian processes and characterized by arbitrary marginal copula distributions. First of all, we establish limit theorems for the marginal and quantile marginal empirical processes of multivariate stationary long-range dependent sequences under Gaussian subordination. Furthermore, we establish the asymptotic behavior of sequential empirical copula processes under non-restrictive smoothness assumptions. The limiting processes in the case of lo…
We study a broad class of asymmetric copulas introduced by Liebscher (2008) as a combination of multiple - usually symmetric - copulas. The main thrust of the paper is to provide new theoretical properties including exact tail dependence expressions and stability properties. A subclass of Liebscher copulas obtained by combining Fréchet copulas is studied in more details. We establish further dependence properties for copulas of this class and show that they are characterized by an arbitrary number of singular components. Furthermore, we introduce a novel iterative representation for general Li…
Regular vine copulas can describe a wider array of dependency patterns than the multivariate Gaussian copula or the multivariate Student's t copula. This paper presents two contributions related to model selection of regular vine copulas. First, our pair copula family selection procedure extends existing Bayesian family selection methods by allowing pair families to be chosen from an arbitrary set of candidate families. Second, our method represents the first Bayesian model selection approach to include the regular vine density construction in its scope of inference. The merits of our approach…
In many applications including financial risk measurement, copulas have shown to be a powerful building block to reflect multivariate dependence between several random variables including the mapping of tail dependencies. A famous key result in this field is Sklar's Theorem. Meanwhile, there exist several approaches to prove Sklar's Theorem in its full generality. An elegant probabilistic proof was provided by L. Rüschendorf. To this end he implemented a certain "distributional transform" which naturally transforms an arbitrary distribution function $F$ to a flexible parameter-dependent funct…
Identifiable Bayesian Deep Generative Copulas with Unknown Layer Widths for Data with Arbitrary Marginal Distributions
Department of Statistics and Data Science（统计与数据科学系）；Washington University in St. Louis（圣路易斯华盛顿大学）；Department of Statistics（统计系）；Columbia University（哥伦比亚大学）
We introduce a copula mixture model to perform dependency-seeking clustering when co-occurring samples from different data sources are available. The model takes advantage of the great flexibility offered by the copulas framework to extend mixtures of Canonical Correlation Analysis to multivariate data with arbitrary continuous marginal densities. We formulate our model as a non-parametric Bayesian mixture, while providing efficient MCMC inference. Experiments on synthetic and real data demonstrate that the increased flexibility of the copula mixture significantly improves the clustering and t…
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 e…
Vine copulas allow to build flexible dependence models for an arbitrary number of variables using only bivariate building blocks. The number of parameters in a vine copula model increases quadratically with the dimension, which poses new challenges in high-dimensional applications. To alleviate the computational burden and risk of overfitting, we propose a modified Bayesian information criterion (BIC) tailored to sparse vine copula models. We show that the criterion can consistently distinguish between the true and alternative models under less stringent conditions than the classical BIC. The …
Pair-copula Bayesian networks (PCBNs) are a novel class of multivariate statistical models, which combine the distributional flexibility of pair-copula constructions (PCCs) with the parsimony of conditional independence models associated with directed acyclic graphs (DAG). We are first to provide generic algorithms for random sampling and likelihood inference in arbitrary PCBNs as well as for selecting orderings of the parents of the vertices in the underlying graphs. Model selection of the DAG is facilitated using a version of the well-known PC algorithm which is based on a novel test for con…
Dupuytren disease is a fibroproliferative disorder with unknown etiology that often progresses and eventually can cause permanent contractures of the affected fingers. In this paper, we provide a computationally efficient Bayesian framework to discover potential risk factors and investigate which fingers are jointly affected. Our Bayesian approach is based on Gaussian copula graphical models, which are one potential way to discover the underlying conditional independence structure of variables in multivariate mixed data. In particular, we combine the semiparametric Gaussian copula with extende…
The cumulative distribution network (CDN) is a recently developed class of probabilistic graphical models (PGMs) permitting a copula factorization, in which the CDF, rather than the density, is factored. Despite there being much recent interest within the machine learning community about copula representations, there has been scarce research into the CDN, its amalgamation with copula theory, and no evaluation of its performance. Algorithms for inference, sampling, and learning in these models are underdeveloped compared those of other PGMs, hindering widerspread use. One advantage of the CDN i…
Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an iid random sample from a multivariate distribution with known margins and unknown extreme-value copula, an extension of the Capéraà-Fougères-Genest estimator was introduced by D. Zhang, M. T. Wells and L. Peng [Journal of Multivariate Analysis 99 (2008) 577-588]. The joint asymptotic distribution of the estimator as a random function on the simplex was not provided. Moreover, implementation of …
The multivariate Hilbert-Schmidt-Independence-Criterion (dHSIC) and distance multivariance allow to measure and test independence of an arbitrary number of random vectors with arbitrary dimensions. Here we define versions which only depend on an underlying copula. The approach is based on the distributional transform, yielding dependence measures which always feature a natural invariance with respect to scalings and translations. Moreover, it requires no distributional assumptions, i.e., the distributions can be of pure type or any mixture of discrete and continuous distributions and (in our s…
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, …
Reproducibility is essential to reliable scientific discovery in high-throughput experiments. In this work we propose a unified approach to measure the reproducibility of findings identified from replicate experiments and identify putative discoveries using reproducibility. Unlike the usual scalar measures of reproducibility, our approach creates a curve, which quantitatively assesses when the findings are no longer consistent across replicates. Our curve is fitted by a copula mixture model, from which we derive a quantitative reproducibility score, which we call the "irreproducible discovery …
