Workshop on copulas
September 13, 2016
Location : Universite catholique de Louvain
Socrate -242, Bâtiment Michotte, place du Cardinal Mercier 10, Louvain-la-Neuve.
Program:
13h30-14h30 : Roger Nelsen (Lewis & Clark College, Portland, Oregon)
14h30-15h00 : Roel Braekers (Universiteit Hasselt)
15h00-15h30 : coffee break
15h30-16h00 : Anouar El Ghouch (Université catholique de Louvain)
16h00-17h00 : Irène Gijbels (Katholieke Universiteit Leuven)
Please note that registration is free but compulsory
Abstracts
Roger Nelsen
"The Structure of the Class of Maximum Entropy Copulas"
A maximum entropy copula is the copula associated with the joint distribution, with prescribed marginal distributions on [0,1], which maximizes the Tsallis-Havrda-Chavát entropy with q = 2. We find necessary and sufficient conditions for each maximum entropy copula to be a copula in the class introduced in Rodríguez-Lallena & Úbeda-Flores (2004), and we also show that each copula in that class is a maximum entropy copula. This is joint work with Verónica González-López and Jesús García.
Roel Braekers
"On the use of copula functions in clustered multivariate survival data of unbalanced size"
In the analysis of clustered multivariate survival data, two different types of models are commonly used if we take the association between the different lifetimes into account: frailty models and copula models. Frailty models assume that, conditional on a common unknown frailty term for each cluster, the hazard function of each individual within that cluster is independent. These unknown frailty terms with their imposed distribution are used to express the association between the different individuals in a cluster. Copula models on the other hand assume that the joint survival function of the individuals within a cluster is given by a copula function, evaluated in the marginal survival function of each individual. Hereby it is the copula function which describes the association between the lifetimes within a cluster. Due to the censoring in the data, copula models have a major disadvantage over frailty models in the sense that the size of the different clusters must be small and equally balanced in order to set up manageable estimation procedures for the different parameters in the copula model. We describe in this talk the use of a copula model in the analysis of unbalanced clustered survival data by focusing on specific classes of copula functions. Hereby we also specifically look at the class of Archimedean copula functions with completely monotone generators and exploit the Laplace transform-expression of these generators to simplify the likelihood function. A results, we develop for this model, one- and two-stage procedures to estimate the association parameter for the copula function. In the one-stage procedure, we consider a parametric model for the marginal survival functions while in the two-stage procedure we model the marginal survival functions by either a parametric or a non-parametric model. As results, we show the consistency and asymptotic normality of the estimator for the association parameters. We perform a simulation study to investigate the finite sample properties of this estimator and finally we illustrate this copula model on a real life data set in which we study the time until first insemination for cows which are clustered within different herds.
Anouar El Ghouch
"Semiparametric copula quantile regression for complete or censored data"
When facing multivariate covariates, general semiparametric regression techniques come at hand to propose flexible models that are unexposed to the curse of dimensionality. A semiparametric copula-based estimator for conditional quantiles is investigated for complete or right-censored data. Extending recent work, the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to an easily implementable estimator expressed, for both contexts with or without censoring, as a weighted quantile of the observed response variable. In addition, and contrary to the initial suggestion in the literature, a semiparametric estimation scheme for the multivariate copula density is studied, motivated by the possible shortcomings of a purely parametric approach and driven by the regression context. The resulting quantile regression estimator has the valuable property of being automatically monotonic across quantile levels, and asymptotic normality for both complete and censored data is obtained under classical regularity conditions. Finally, numerical examples as well as a real data application are used to illustrate the validity and finite sample performance of the proposed procedure.
Irène Gijbels
"Testing for no covariate effect in conditional copulas"
In dependence modelling using conditional copulas, one often imposes the working assumption that the covariate(s) influences the conditional copula solely through the marginal distributions. This so-called (pairwise) simplifying assumption is, for example, often made in vine copula constructions. In this talk the main interest is in testing whether this assumption holds or not. We propose some nonparametric tests for testing the null hypothesis of the simplifying assumption, and study their asymptotic behaviours, under the null hypothesis and under some local alternatives. The emphasis is on tests which are fully nonparametric in nature: not requiring choices of copula families nor knowledge of the marginals. The finite-sample performances of the discussed tests are investigated via a simulation study. A real data application illustrates the use of the tests. This talk is based on joint work with M. Omelka and N. Veraverbeke.