  Statistical inference can be performed by minimizing, over the parameter
space, the Wasserstein distance between model distributions and the empirical
distribution of the data. We study asymptotic properties of such minimum
Wasserstein distance estimators, complementing results derived by Bassetti,
Bodini and Regazzini in 2006. In particular, our results cover the misspecified
setting, in which the data-generating process is not assumed to be part of the
family of distributions described by the model. Our results are motivated by
recent applications of minimum Wasserstein estimators to complex generative
models. We discuss some difficulties arising in the approximation of these
estimators and illustrate their behavior in several numerical experiments. Two
of our examples are taken from the literature on approximate Bayesian
computation and have likelihood functions that are not analytically tractable.
Two other examples involve misspecified models.
