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Rank-based inference for bivariate extreme-value copulas. (English) Zbl 1173.62013

Summary: Consider a continuous random pair \((X,Y)\) whose dependence is characterized by an extreme-value copula with Pickands dependence function \(A\). When the marginal distributions of \(X\) and \(Y\) are known, several consistent estimators of \(A\) are available. Most of them are variants of the estimators due to J. Pickands [Bull. Int. Stat. Inst. 49, 859–878 (1981; Zbl 0518.62045)] and P. Capéraà, A.-L. Fougères and C. Genest [Biometrika 84, No. 3, 567–577 (1997; Zbl 1058.62516)]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of \(X\) and \(Y\) are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given.

MSC:

62G05 Nonparametric estimation
62G32 Statistics of extreme values; tail inference
62G20 Asymptotic properties of nonparametric inference
65C05 Monte Carlo methods
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