Estimation of a copula when a covariate affects only marginal distributions

Abstract

This paper is concerned with studying the dependence structure between two random variables Y1 and Y2 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 Y1 and Y2 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 p on the marginals of Y1 and Y2 and show that in these cases, weak convergence of the estimator, at n-rate, holds. The theoretical results are illustrated by simulations and a real data example.