Abstract
Abstract Genest and Segers (2010) gave conditions under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance than the standard empirical process based on a random sample from the underlying copula. An extension of this result to the multivariate case is provided.
Highlights
Let X, . . . , Xd be random variables with joint cumulative distribution function H and continuous univariate margins F, . . . , Fd, respectively
Genest and Segers (2010) gave conditions under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance than the standard empirical process based on a random sample from the underlying copula
As a consequence of Lemma 2, a d-variate Archimedean copula with a completely monotone generator ψ is necessarily left-tail decreasing (LTD)-VV given that ψ is log-convex in this case; see the proof of Corollary 4.6.3 in [20]
Summary
Let X , . . . , Xd be random variables with joint cumulative distribution function H and continuous univariate margins F , . . . , Fd, respectively. These authors do not go beyond the bivariate case, except to state that their result holds for arbitrary d ≥ when C is the independence copula de ned, for all u , . This observation is used in the following example to show, among other things, that Theorem 1 extends Proposition 4 of Genest and Segers [11] stating that inequality (4) holds for the independence copula Πd.
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