Generalized diagonal band copulas

This paper investigates the class of generalized diagonal band (GDB) copulas, which form an extension of diagonal band (DB) copulas originally introduced by Cooke and Waij [Cooke, R.N., Waij, R., 1986. Monte Carlo sampling for generalized knowledge dependence, with application to human reliability. Risk Anal. 6, 335-343] for application in simulating high-dimensional distributions for dependent random variables. Ferguson [Ferguson, T.S., 1995. A class of bivariate uniform distributions. Stat. Pap. 36, 31-40] provided a method for constructing copulas which include the convex closure of DB copulas. Bojarski [Bojarski, J., 2001. A New Class of Band Copulas-Distributions with Uniform Marginals, Technical Publication. Institute of Mathematics, Technical University of Zielona Góra] provided a different extension of the original DB construction and obtained (independently) the same class as Ferguson, which we now call the GDB. Our interest in GDB copulas results from their possible application in the vine-copula method for simulating high-dimensional dependent vectors; see Cooke [Cooke, R.M., 1997. Markov and entropy properties of tree- and vine-dependent variables. Proceedings of the ASA Section on Bayesian Statistical Science], Bedford and Cooke [Ann. Math. Artif. Intel. 32 (2001) 245-268], and Kurowicka and Cooke [Linear Algebra Appl. 372 (2003) 225-251]. This method requires using copulas with certain properties which GDB copulas often possess. We list these requirements and construct distributions that meet them using the class of GDB copulas.