  Data consisting of samples of probability density functions are increasingly
prevalent, necessitating the development of methodologies for their analysis
that respect the inherent nonlinearities associated with densities. In many
applications, density curves appear as functional response objects in a
regression model with vector predictors. For such models, inference is key to
understand the importance of density-predictor relationships, and the
uncertainty associated with the estimated conditional mean densities, defined
as conditional Fr\'echet means under a suitable metric. Using the Wasserstein
geometry of optimal transport, we consider the Fr\'echet regression of density
curve responses and develop tests for global and partial effects, as well as
simultaneous confidence bands for estimated conditional mean densities. The
asymptotic behavior of these objects is based on underlying functional central
limit theorems within Wasserstein space, and we demonstrate that they are
asymptotically of the correct size and coverage, with uniformly strong
consistency of the proposed tests under sequences of contiguous alternatives.
The accuracy of these methods, including nominal size, power, and coverage, is
assessed through simulations, and their utility is illustrated through a
regression analysis of post-intracerebral hemorrhage hematoma densities and
their associations with a set of clinical and radiological covariates.
