A family of transformed copulas with singular component

TL;DR

This paper introduces a new family of bivariate copulas created by transforming existing copulas with increasing functions, highlighting their singular components and tail dependence properties, with implications for dependence modeling.

Contribution

It proposes a novel class of transformed copulas with singular components, providing conditions for their validity and analyzing their dependence properties.

Findings

Identified conditions for transformed functions to be valid copulas.

Demonstrated the existence of singular components along the main diagonal.

Derived tail dependence coefficients for the new copulas.

Abstract

In this paper, we present a family of bivariate copulas by transforming a given copula function with two increasing functions, named as transformed copula. One distinctive characteristic of the transformed copula is its singular component along the main diagonal. Conditions guaranteeing the transformed function to be a copula function are provided, and several classes of the transformed copulas are given. The singular component along the main diagonal of the transformed copula is verified, and the tail dependence coefficients of the transformed copulas are obtained. Finally, some properties of the transformed copula are discussed, such as the totally positive of order 2 and the concordance order.