Constructing Copulas by Means of Pairs of Order Statistics

The construction of distributions with given marginals has been a problem of interest to statisticians for many years. Today, in view of Sklar’s theorem [25, 26], this problem can be reduced to the construction of a copula. Copulas are a special type of aggregation functions, and nowadays are of interest in fuzzy set theory [5, 6, 12, 14, 16]. Nelsen [22] summarizes different methods of constructing copulas. Recently, various authors provide construction methods from the class of copulas to itself, or from a more general class of functions to another (see, for instance, [2, 8, 10, 11, 13, 18, 19, 20]). One of the purposes for such constructions is to increase the availability of copulas for modeling purposes. In this paper we provide two new transformations of copulas, which recover some known families of copulas. The method is based on the choice of pairs of order statistics of the marginal distributions. Properties on dependence, symmetry, invariance, and random number generation of such transformations are also shown. Preliminary ideas can be found in [3].