Weak convergence of the weighted empirical beta copula process

Abstract

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.

Subsampling (weighted smooth) empirical copula processes

“…A first seminal result in that direction is due to Berghaus et al [1] for the empirical copula processes C n andC n . Berghaus and Segers [2] have shown a similar result for the empirical beta copula process C β n . Because the latter involves the empirical beta copula C β n , which is a genuine copula, its statement is simpler and takes the form of a weighted weak convergence in ∞ ([0, 1] d ) (that is, with respect to the uniform metric).…”

“…Note that Condition 4.2 first appeared in [44] where it was used it to prove the almost sure representation for C n originally conjectured in [49]. As discussed in [2], this condition is satisfied for several commonly occurring copulas. Theorem 2 in [2] then states that, under Conditions 2.1, 2.4, 4.1 and 4.2, for any ω ∈ [0, 1/2), the weighted empirical beta copula process C β n /g ω converges weakly in ∞ ([0, 1] d ) to C C /g ω , where C C and g are given in (2.18) and (4.1), respectively.…”

“…Using the fact that C # n is a copula almost surely, we can proceed exactly as in the proof of Lemma 8 in [2] to show that, almost surely, sup…”

“…. , F d are continuous: as suggested in [2], take, for instance, a Markov chain where the current state is repeated with positive probability.…”

An estimator of the stable tail dependence function based on the empirical beta copula

“…Einmahl et al (2006) proved weak convergence of the (bivariate) empirical stdf with respect to the topology of a weighted supremum norm, providing more information on the estimator in a neighbourhood of the origin. Similarly, and Berghaus and Segers (2017) established weighted weak convergence for the empirical copula and empirical beta copula, respectively. It is an open problem whether such weighted weak convergence also holds for the empirical stdf and empirical beta stdf in arbitrary dimensions.…”