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Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. (English) Zbl 1243.62066
Summary: Weak convergence of an empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in W. Stute’s [Ann. probab. 12, 361–379 (1984; Zbl 0533.62037)] representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of Gaussian copulas with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.

MSC:
62G30 Order statistics; empirical distribution functions
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G20 Asymptotic properties of nonparametric inference
60F15 Strong limit theorems
60F05 Central limit and other weak theorems
62G32 Statistics of extreme values; tail inference
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TwoCop
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