  Wasserstein geometry and information geometry are two important structures to
be introduced in a manifold of probability distributions. Wasserstein geometry
is defined by using the transportation cost between two distributions, so it
reflects the metric of the base manifold on which the distributions are
defined. Information geometry is defined to be invariant under reversible
transformations of the base space. Both have their own merits for applications.
In particular, statistical inference is based upon information geometry, where
the Fisher metric plays a fundamental role, whereas Wasserstein geometry is
useful in computer vision and AI applications. In this study, we analyze
statistical inference based on the Wasserstein geometry in the case that the
base space is one-dimensional. By using the location-scale model, we further
derive the W-estimator that explicitly minimizes the transportation cost from
the empirical distribution to a statistical model and study its asymptotic
behaviors. We show that the W-estimator is consistent and explicitly give its
asymptotic distribution by using the functional delta method. The W-estimator
is Fisher efficient in the Gaussian case.
