Statistica Sinica 7(1997), 713-738

NON-LINEAR INTEGRAL EQUATIONS TO APPROXIMATE

BIVARIATE DENSITIES WITH GIVEN MARGINALS

AND DEPENDENCE FUNCTION

Geert Molenberghs and Emmanuel Lesaffre

Limburgs Universitair Centrum and Katholieke Universiteit Leuven

Abstract: The local dependence function, introduced by Holland and Wang (1987) and studied by Wang (1993) as a continuous version of the local cross-ratio, describes the local relation between two random variables. Three explicit numerical algorithms are proposed to approximate bivariate densities given the marginal densities and the local dependence function. This approach is suited for simulation purposes, to provide illustrative examples of densities with given marginals, and for estimation of model parameters. The technique involves non-classical integral equation theory. The accuracy of the approximations is investigated.

Key words and phrases: Bivariate density, integral equation, local cross-ratio, local dependence function, numerical integration, Plackett family.