We consider the copula mapping, which maps a joint cumulative distribution function to the corresponding copula. Its Hadamard differentiablity was shown in van der Vaart and Wellner (1996), Fermanian et al. (2004) and (under less strict assumptions) in B\"ucher and Volgushev (2013). This differentiability result has proved to be a powerful tool to show weak convergence of empirical copula processes in various settings using the functional delta method. We state a generalization of the Hadamard differentiability results that simplifies the derivations of asymptotic expansions and weak convergence of empirical copula processes in the presence of covariates. The usefulness of this result is illustrated on several applications which include a multidimensional functional linear model, where the copula of the error vector describes the dependency between the components of the vector of observations, given the functional covariate.
翻译：我们考虑构拟映射，它将联合累积分布函数映射到相应的构拟。van der Vaart和Wellner（1996），Fermanian等人（2004）和B\"ucher和Volgushev（2013）证明了它的Hadamard可微性。该可微性结果已经证明是一个强大的工具，可利用函数$\delta$方法，在各种设置中显示经验构拟过程的弱收敛性。我们阐述了Hadamard可微性结论的一个普遍化结果，在协变量存在的情况下简化了渐近展开和经验构拟过程的弱收敛性的推导。我们通过多维函数线性模型来阐明这一结果的实用性，其中误差向量的构拟描述了观察向量的分量之间的依赖关系，给定函数协变量。
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