Smooth semi- and nonparametric bayesian estimation of bivariate densities from bivariate histogram data

We show how penalized B-splines combined with the composite link model can be used to estimate a bivariate density from histogram data. Two strategies are proposed: the first one is semi-parametric with exible margins modeled using B-splines and a parametric copula for the dependence structure ; the second one is nonparametric and is based on Kronecker products of the marginal B-splines bases. Frequentist and Bayesian estimations are described. A large simulation study quantifies the performances of both methods under dierent dependence structures and varying strengths of dependence, sample sizes and amounts of grouping. It suggests that Schwarz's BIC is a good tool for classifying the competing models. The density estimates are used to evaluate conditional quantiles in two applications in social and in medical sciences.