Conditional copulas, association measures and their applications

Authors:
Irène Gijbels;Noël Veraverbeke;Marel Omelka
Affiliations:
Department of Mathematics and Leuven Statistics Research Center (LStat), Katholieke Universiteit Leuven, Celestijnenlaan 200B, Box 2400, B-3001 Leuven (Heverlee), Belgium;Center for Statistics, Hasselt University, Agoralaan-building D, B-3590 Diepenbeek, Belgium;Jaroslav Hájek Center for Theoretical and Applied Statistics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic
Venue:
Computational Statistics & Data Analysis
Year:
2011

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Abstract

One way to model a dependence structure is through the copula function which is a mean to capture the dependence structure in the joint distribution of variables. Association measures such as Kendall's tau or Spearman's rho can be expressed as functionals of the copula. The dependence structure between two variables can be highly influenced by a covariate, and it is of real interest to know how this dependence structure changes with the value taken by the covariate. This motivates the need for introducing conditional copulas, and the associated conditional Kendall's tau and Spearman's rho association measures. After the introduction and motivation of these concepts, two nonparametric estimators for a conditional copula are proposed and discussed. Then nonparametric estimates for the conditional association measures are derived. A key issue is that these measures are now looked at as functions in the covariate. The performances of all estimators are investigated via a simulation study which also includes a data-driven algorithm for choosing the smoothing parameters. The usefulness of the methods is illustrated on two real data examples.