Construction of bivariate S-distributions with copulas

S-distributions are univariate statistical distributions with four parameters. They have a simple mathematical structure yet provide excellent approximations for many traditional distributions and also contain a multitude of distributional shapes without a traditional analog. S-distributions furthermore have a number of beneficial features, for instance, in terms of data classification and scaling properties. They provide an appealing compromise between generality in data representation and logistic simplicity and have been applied in a variety of fields from applied biostatistics to survival analysis and risk assessment. Given their advantages in the single- variable case, it is desirable to extend S-distributions to several variates. This article proposes such an extension. It focuses on bivariate distributions whose marginals are S-distributions, but it is clear how more than two variates are to be addressed. The construction of bivariate S- distributions utilizes copulas, which have been developed quite rapidly in recent years. It is demonstrated here how one may generate such copulas and employ them to construct and analyze bivariate—and, by extension, multivariate—S-distributions. Particular emphasis is placed on Archimedean copulas, because they are easy to implement, yet quite flexible in fitting a variety of distributional shapes. It is illustrated that the bivariate S-distributions thus constructed have considerable flexibility. They cover a variety of marginals and a wide range of dependences between the variates and facilitate the formulation of relationships between measures of dependence and model parameters. Several examples of marginals and copulas illustrate the flexibility of bivariate S-distributions.