Nearly Nonparametric Multivariate Density Estimates That Incorporate Marginal Parametric Density Information

When data analysts have multivariate data, often they have partial knowledge about the form of the marginal densities, but frequently they have little information about the bivariate and higher dimensional densities. This article provides nonparametric estimators that nearly equal the MLE estimates for the marginal densities while being close to the kernel nonparametric density estimates for the joint density estimates, provided that the assumption about the marginal densities is correct. The motivation for this article came from recollections of a 15-year-old conversation with Ingram Olkin where the problem at hand was how to model multivariate data with fixed marginals but with a flexible and rich multivariate structure.