## Bernoulli

• Bernoulli
• Volume 23, Number 1 (2017), 743-772.

### Weak convergence of the empirical copula process with respect to weighted metrics

#### Abstract

The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are allowed to be serially dependent) with respect to weighted supremum distances. The usefulness of our results is illustrated by applications to general bivariate rank statistics and to estimation procedures for the Pickands dependence function arising in multivariate extreme-value theory.

#### Article information

Source
Bernoulli, Volume 23, Number 1 (2017), 743-772.

Dates
Revised: June 2015
First available in Project Euclid: 27 September 2016

https://projecteuclid.org/euclid.bj/1475001373

Digital Object Identifier
doi:10.3150/15-BEJ751

Mathematical Reviews number (MathSciNet)
MR3556791

Zentralblatt MATH identifier
1367.60026

#### Citation

Berghaus, Betina; Bücher, Axel; Volgushev, Stanislav. Weak convergence of the empirical copula process with respect to weighted metrics. Bernoulli 23 (2017), no. 1, 743--772. doi:10.3150/15-BEJ751. https://projecteuclid.org/euclid.bj/1475001373

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#### Supplemental materials

• Supplement to “Weak convergence of the empirical copula process with respect to weighted metrics”. A detailed exposition on bounded variation and Lebesgue–Stieltjes integration for two-variate functions and the proofs of Proposition 4.4 and of Lemma 4.9 can be found in [3].