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Weak convergence of the weighted empirical beta copula process. (English) Zbl 1404.62049
The authors of this paper study asymptotic properties of empirical beta copulas. Since the empirical beta copula is itself a copula, it was possible to prove weighted weak convergence for the empirical beta copula process on the whole unit cube, being the main result of the paper. Weak convergence on the whole unit cube rather than on a subset thereof is quite handy since it allows for a direct application of, e.g., the continuous mapping theorem. In particular, there is no longer any need to treat the boundary regions separately. Two applications are considered. First, the Cramér-von Mises test statistic for independence studied by C. Genest and B. Rémillard [Test 13, No. 2, 335–370 (2004; Zbl 1069.62039)] is modified using the empirical beta copula and adding a weight function in the integral, emphasizing the tails. The asymptotic distribution of the statistic under the null hypothesis is then a corollary of the main result of this paper. A second considered application is the Capéraà-Fougères-Genest estimator [P. Capéraà et al., Biometrika 84, No. 3, 567–577 (1997; Zbl 1058.62516)] of the Pickands dependence function of a multivariate extreme-value copula. Under weak dependence, replacing the empirical copula by the empirical beta copula yields a more accurate estimator. Its asymptotic distribution is again an immediate consequence of the main result of this paper.

62G32 Statistics of extreme values; tail inference
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60F05 Central limit and other weak theorems
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
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