Aggregation functions constructed from copulas and semi-copulas

Abstract

In this dissertation, we show that the set of aggregation functions constructed by composing semi-copulas and tuples of non-decreasing univariate functions is dense in the set of all aggregation functions. In particular, this construction method works for all strictly increasing aggregation functions. This method also resembles Sklar’s construction of multivariate distribution functions. In addition, we obtain necessary and sufficient conditions for representing aggregation functions in a form of our method. Furthermore, we show that any smooth semi-copulas can be constructed from copulas. Therefore, we can also use copulas and univariate non-decreasing functions to construct (continuous) aggregation functions. This allows us to reuse abundant copula families to construct new aggregation functions. Moreover, we prove that the set of aggregation functions constructed by this method is dense in the set of continuous aggregation functions.