Rank-Based Inference for Bivariate Extreme-Value Copulas

Several nonparametric estimators are available for the Pickands dependence function of a bivariate extreme-value copula. All of them, however, require knowledge of the univariate margins. In this paper, rank-based versions of some of these estimators are proposed for the case where the margins are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to determine the asymptotic distributions of rank-based versions of the estimator of Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878], some of its variants, and the estimator of Cap\'era\`a, Foug\`eres and Genest [Biometrika 84 (1997) 567--577]. At independence, the latter estimator is shown to be asymptotically more efficient than its competitors. Simulations indicate that this remains valid in small samples and for other extreme-value copulas. In addition, the use of ranks is found to improve the performance of all estimators considered, even when the margins are known.