Journal of the American Statistical Association · 1993 · 250 citations · 0 references
Natural SciencesMathematicsProbability TheoryStatistical ModelingStatisticsDependence ModelingFrailty ModelsInferential StatisticsJoint DistributionsMultivariate AnalysisStatistical InferenceArchimedean CopulasAsymptotic PropertiesBivariate DistributionsCopula EstimationCopula ModelsDependence StructureEmpirical Distribution
Archimedean copulas provide a flexible framework for modeling bivariate dependence, encompassing families such as Gumbel, Clayton, Frank, and others, and are also used in frailty models. This article investigates how to select an appropriate Archimedean copula to represent the dependence between two variables based on a random sample. The authors develop a semiparametric estimator that uses a one‑dimensional empirical distribution function and a decomposition of Kendall’s tau, allowing estimation without specifying marginal distributions. The estimator is √n‑consistent with a closed‑form asymptotic variance, enabling a data‑driven strategy for choosing the best Archimedean family, as demonstrated on a uranium exploration dataset. The authors note that the method could be extended to multivariate or censored data settings.
Abstract A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = ϕ–1[ϕ{F(x)} + ϕ{G(y)}] for some convex, decreasing function ϕ defined on [0, 1] in such a way that ϕ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thélot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X 1, Y 1), …, (X n , Y n ). The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic, is seen to be √n-consistent, and an explicit formula for its asymptotic variance is provided. This leads to a strategy for selecting the parametric family of Archimedean copulas that provides the best possible fit to a given set of data. To illustrate these procedures, a uranium exploration data set is reanalyzed. Although the presentation is restricted to problems involving a random sample from a bivariate distribution, extensions to situations involving multivariate or censored data could be envisaged.