STATISTICAL-INFERENCE PROCEDURES FOR BIVARIATE ARCHIMEDEAN COPULAS

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 express ed in the form H(x, y) = phi-1[phi{F(x)} + phi{G(y)}] for some convex, decreasing function phi defined on (0, 1] in such a way that phi(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thelot, Clayton, Fra nk, and Hougaard. Frailty models also fall under that general prescrip tion. This article examines the problem of selecting an Archimedean co pula providing a suitable representation of the dependence structure b etween two variates X and Y in the light of a random sample (X1, Y1),. .., (X(n), Y(n)). The key to the estimation procedure is a one-dimensi onal empirical distribution function that can be constructed whether t he uniform representation of X and Y is Archimedean or not, and indepe ndently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic, is seen to be square-root n- consistent, and an explicit formula for its asymptotic variance is pro vided. This leads to a strategy for selecting the parametric family of Archimedean copulas that provides the best possible fit to a given se t 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 t o situations involving multivariate or censored data could be envisage d.