Estimation of a Copula when a Covariate Affects only Marginal Distributions

摘要

This paper is concerned with studying the dependence structure between two random variables Y-1 and Y-2 in the presence of a covariate X, which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y-1 and Y-2 given X, does not depend on the value of X. This latter independence often appears as a simplifying assumption in pair-copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location-scale models for the effect of the covariate X on the marginals of Y-1 and Y-2 and show that in these cases, weak convergence of the estimator, at root n-rate, holds. The theoretical results are illustrated by simulations and a real data example.