Bivariate Copula Transformations Based on Rigid Motions and Distortions

The Copula function is used to describe the relationship between random variables. Through various copulas, we can adapt various models for data analysis. In this respect, a copula family with a flexible structure can help to analyze data with various structures. Rigid motion (Fuchs and Schmidt, 2014) and asymmetric transformation (Khoudraji, 1995) for the copulas are the representative achievements for creating flexible copula families. In this paper, we propose a new method to extend the given bivariate copula family by combining asymmetric transformation and rigid motions. Using the basic group theory, we classify the set of newly transformed copulas and study the related properties focusing on the concordance order. We estimate Stock prices of Goldman Sachs and JP Morgan from January 2010 to July 2017 using the proposed copulas.