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.

The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both a disjoint and a sliding blocks estimator for the extremal index are analyzed in detail. In contrast to many competitors, the estimators only depend on the choice of one parameter sequence. We derive an asymptotic expansion, prove asymptotic normality and show consistency of an estimator for the asymptotic variance. Explicit calculations in certain models and a finite-sample Monte Carlo simulation study reveal that the sliding blocks estimator outperforms other blocks estimators, and that it is competitive to runs-and inter-exceedance estimators in various models. The methods are applied to a variety of financial time series.

The empirical copula has proved to be useful in the construction and understanding of many statistical procedures related to dependence within random vectors. The empirical beta copula is a smoothed version of the empirical copula that enjoys better finite-sample properties. At the core lie fundamental results on the weak convergence of the empirical copula and empirical beta copula processes. Their scope of application can be increased by considering weighted versions of these processes. In this paper we show weak convergence for the weighted empirical beta copula process. The weak convergence result for the weighted empirical beta copula process is stronger than the one for the empirical copula and its use is more straightforward. The simplicity of its application is illustrated for weighted Cramér-von Mises tests for independence and for the estimation of the Pickands dependence function of an extremevalue copula.

In recent years, stationary time series models based on copula functions became increasingly popular in econometrics to model nonlinear temporal and cross-sectional dependencies. Within these models, we consider the problem of testing the goodness-of-fit of the parametric form of the underlying copula. Our approach is based on a dependent multiplier bootstrap and it can be applied to any stationary, strongly mixing time series. The method extends recent i.i.d. results by Kojadinovic, Yan and Holmes [I. Kojadinovic, Y. Yan and M. Holmes, Fast large sample goodness-of-fit tests for copulas, Statistica Sinica 21 (2011), 841-871] and shares the same computational benefits compared to methods based on a parametric bootstrap. The finite-sample performance of our approach is investigated by Monte Carlo experiments for the case of copula-based Markovian time series models.

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