A Family of Bivariate Densities Constructed from Marginals

Abstract

A new family of bivariate densities with specified marginal densities is described. The bivariate dependence structure takes the form of a density weighting function constructed as follows. The space of continuous variates X and Y is mapped into a unit square through probability integral transformations. A polygonal partition of the unit square supports a bivariate covariance characteristic, which is structured as a functional of a univariate regression characteristic. This structure (a) exhibits all geometric features of positive (or negative) dependence implied by Fréchet bounds, which it can attain, and (b) prescribes positive (or negative) mutual regression dependence between X and Y. To obtain a parametric family of bivariate densities, the regression characteristic is given a power form. The resultant two-parameter model has specified marginals and, within certain constraints, allows one to separately control the shape of the bivariate density and the degree of association between X and Y; their Spearman's correlation coefficient p can attain any value: −1 < p < 1.

Key Words:

• Density weighting function
• Dependence structure
• Fréchet bounds
• Nonlinear regression
• Probability integral transformation
• Regression dependence