Absolutely continuous copulas obtained by regularization of the Frech\'{e}t--Hoeffding bounds

TL;DR

This paper presents a constructive method to regularize the singular lower and upper Frechét-Hoeffding copulas into absolutely continuous copulas through averaging, providing explicit conditions for this transformation.

Contribution

It introduces a new explicit and constructive regularization technique for transforming singular copulas into absolutely continuous ones.

Findings

Regularization of singular copulas is achievable via averaging.

The method provides explicit sufficient conditions for regularization.

Higher degrees of regularization are not attainable with this approach.

Abstract

We show that the lower and upper Frech\'{e}t-Hoeffding copulas, which are singular, can be regularized to absolutely continuous copulas. The method, which is constructive and explicit, states sufficient conditions for when an absolutely continuous copula can be achieved by averaging. A higher degree of regularisation cannot be achieved with the proposed method.