A note on Bertino and Fredricks–Nelsen copulas

Bertino and Fredricks–Nelsen copulas are the most prominent examples of copulas which are constructed from a diagonal. For a given diagonal, it is known that the Bertino copula is dominated by the Fredricks–Nelsen copula. In this paper, we show that these copulas are either distinct or both equal to the upper Fréchet–Hoeffding bound. We also prove known and new representations of Spearman’s rho, Gini’s gamma and Kendall’s tau for Bertino and Fredricks–Nelsen copulas in terms of their diagonal, and we show that, under any convex–linear measure of concordance or Kendall’s tau, the range of the collection of all copulas having the same diagonal is bounded by the values of the respective Bertino and Fredricks–Nelsen copulas.