Quadratic transformations of multivariate quasi-copula and multivariate semi-copula

Abstract

In this dissertation, we review transformations of a bivariate semi-copula, quasicopula and copula. From the concept of the transformations, we present a construction of a semi-copula (quasi-copula) via a composition of a multivariate semi-copula (quasicopula) with a quadratic polynomial. Actually, such compositions will not always result in a semi-copula (quasi-copula). The aim of this dissertation is to find the necessary and sufficient conditions of the quadratic polynomial for the semi-copula (quasi-copula) transformation. We found that the conditions depend on the coefficients of the quadratic polynomial. Moreover, set of the coefficients forms a convex set with linear boundary. Furthermore, we study a transformation of several multivariate semi-copulas. Besides, the characterization for such transformations are also given.