  The problem of modeling the relationship between univariate distributions and
one or more explanatory variables has found increasing interest. Traditional
functional data methods cannot be applied directly to distributional data
because of their inherent constraints. Modeling distributions as elements of
the Wasserstein space, a geodesic metric space equipped with the Wasserstein
metric that is related to optimal transport, is attractive for statistical
applications. Existing approaches proceed by substituting proxy estimated
distributions for the typically unknown response distributions. These estimates
are obtained from available data but are problematic when for some of the
distributions only few data are available. Such situations are common in
practice and cannot be addressed with available approaches, especially when one
aims at density estimates. We show how this and other problems associated with
density estimation such as tuning parameter selection and bias issues can be
side-stepped when covariates are available. We also introduce a novel version
of distribution-response regression that is based on empirical measures. By
avoiding the preprocessing step of recovering complete individual response
distributions, the proposed approach is applicable when the sample size
available for some of the distributions is small. In this case, one can still
obtain consistent distribution estimates even for distributions with only few
data by gaining strength across the entire sample of distributions, while
traditional approaches where distributions or densities are estimated
individually fail, since sparsely sampled densities cannot be consistently
estimated. The proposed model is demonstrated to outperform existing approaches
through simulations. Its efficacy is corroborated in two case studies on
Environmental Influences on Child Health Outcomes (ECHO) data and eBay auction
data.
