Asymptotic behavior of the empirical multilinear copula process under broad conditions

Abstract

The empirical checkerboard copula is a multilinear extension of the empirical copula, which plays a key role for inference in copula models. Weak convergence of the corresponding empirical process based on a random sample from the underlying multivariate distribution is established here under broad conditions which allow for arbitrary univariate margins. It is only required that the underlying checkerboard copula has continuous first-order partial derivatives on an open subset of the unit hypercube. This assumption is very weak and always satisfied when the margins are discrete. When the margins are continuous, one recovers the limit of the classical empirical copula process under conditions which are comparable to the weakest ones currently available in the literature. A multiplier bootstrap method is also proposed to replicate the limiting process and its validity is established. The empirical checkerboard copula is further shown to be a more precise estimator of the checkerboard copula than the empirical copula based on jittered data. Finally, the weak convergence of the empirical checkerboard copula process is shown to be sufficiently strong to derive the asymptotic behavior of a broad class of functionals that are directly relevant for the development of rigorous statistical methodology for copula models with arbitrary margins.