Mixed data kernel copulas

Abstract

A number of approaches toward the kernel estimation of copula have appeared in the literature. Most existing approaches use a manifestation of the copula that requires kernel density estimation of bounded variates lying on a \(d\)-dimensional unit hypercube. This gives rise to a number of issues as it requires special treatment of the boundary and possible modifications to bandwidth selection routines, among others. Furthermore, existing kernel-based approaches are restricted to continuous data types only, though there is a growing interest in copula estimation with discrete marginals. We demonstrate that using a simple inversion method can sidestep boundary issues while admitting mixed data types directly thereby extending the reach of kernel copula estimators. Bandwidth selection proceeds by a recently proposed cross-validation method. Furthermore, there is no curse of dimensionality for the kernel-based copula estimator (though there is for the copula density estimator, as is the case for existing kernel copula density methods).