On the monotonicity of copula-based conditional distributions

In this paper, we find necessary and sufficient conditions so that copula-based conditional distributions of a response variable with respect to covariates, are ordered with respect to the simple stochastic order introduced by Lehmann. These conditions do not depend on the marginal distributions of the random variables. As a result, we have conditions to ensure that the conditional mean and conditional quantiles are monotonic with respect to the covariates.