Generalized Diagonal Band Copulas with Two-Sided Generating Densities

Copulae are joint continuous distributions with uniform marginals and have been proposed to capture probabilistic dependence between random variables. Maximum entropy copulae introduced by Bedford and Meeuwissen (1997) provide experts the option of making minimally informative assumptions given a degree of dependence constraint between two random variables. Unfortunately, their distributions functions are not available in a closed form, and their application requires the use of numerical methods. In this paper we shall study a sub-family of generalized diagonal band (GDB) copulae, separately introduced by Ferguson (1995) and Bojarski (2001). Similar to Archimedean copulae, GDB copulae construction require a generator function. Bojarski's GDB copula generator functions are symmetric probability density functions. In this paper, members of a symmetric two-sided framework of distributions introduced by Van Dorp and Kotz (2003) shall be considered. This flexible set-up allows for derivations of GDB copula properties resulting in novel convenient expressions. A straightforward elicitation procedure for the GDB copula dependence parameter is proposed. Closed form expressions for specific examples in the sub-family of GDB copulae are presented, which enhance their transparency and facilitate their application. These examples closely approximate the entropy of maximum entropy copulae. Application of GDB copulae is illustrated via a value of information decision analysis example.