  Minimum distance estimation (MDE) gained recent attention as a formulation of
(implicit) generative modeling. It considers minimizing, over model parameters,
a statistical distance between the empirical data distribution and the model.
This formulation lends itself well to theoretical analysis, but typical results
are hindered by the curse of dimensionality. To overcome this and devise a
scalable finite-sample statistical MDE theory, we adopt the framework of smooth
1-Wasserstein distance (SWD) $\mathsf{W}_1^{(\sigma)}$. The SWD was recently
shown to preserve the metric and topological structure of classic Wasserstein
distances, while enjoying dimension-free empirical convergence rates. In this
work, we conduct a thorough statistical study of the minimum smooth Wasserstein
estimators (MSWEs), first proving the estimator's measurability and asymptotic
consistency. We then characterize the limit distribution of the optimal model
parameters and their associated minimal SWD. These results imply an
$O(n^{-1/2})$ generalization bound for generative modeling based on MSWE, which
holds in arbitrary dimension. Our main technical tool is a novel
high-dimensional limit distribution result for empirical
$\mathsf{W}_1^{(\sigma)}$. The characterization of a nondegenerate limit stands
in sharp contrast with the classic empirical 1-Wasserstein distance, for which
a similar result is known only in the one-dimensional case. The validity of our
theory is supported by empirical results, posing the SWD as a potent tool for
learning and inference in high dimensions.
