Median and quantile conditional copulas

This article studies the conditional dependency between random variables, conditionally upon a covariate (vector). The conditional copula fully characterizes this conditional dependency. A way to summarize this dependence structure taking into account the impact of the covariate is via the average conditional copula, which under fairly general conditions coincides with the partial copula. A mean is just one way to summarize this conditional dependence behaviour. In this article, we introduce the notions of median conditional copula and more generally quantile conditional copula. We investigate the existence of these concepts and establish explicit expressions for calculating them. Examples are given to illustrate the concepts, and the practical use of them is demonstrated in real data examples.