Interval-conditioned quantiles of bivariate copulas

Abstract

We study conditional distributions and their quantiles when a continuous covariate is conditioned on an interval event, without discretizing the covariate itself, focusing on bivariate copulas. For strict bivariate Archimedean copulas, we show that interval-conditioned quantiles on an arbitrary bin can be characterized by a one-dimensional equation involving only the copula generator. We identify settings in which numerical integration is unnecessary and provide closed-form expressions for interval-conditioned quantiles in the first (threshold) bin for the Clayton, Frank, Gumbel, and Joe copulas. For bivariate Clayton and Frank copulas, we further establish a unique anchor value inside the bin at which the interval-conditioned quantile equals the pointwise conditional quantile. We further show that last bin formulas can be obtained via the survival copula and derive closed-form last bin quantiles and anchors for the bivariate Frank copula. Numerical illustrations show that equal-probability bins can make distinct copula families appear nearly indistinguishable at central quantiles, whereas tail-favored bins restore separation. Our results offer practical formulas for interval-based inference, which is relevant to privacy-motivated binning.