  Wasserstein geometry and information geometry are two important structures
introduced in a manifold of probability distributions. The former is defined by
using the transportation cost between two distributions, so it reflects the
metric structure of the base manifold on which distributions are defined.
Information geometry is constructed based on the invariance criterion that the
geometry is invariant under reversible transformations of the base space. Both
have their own merits for applications. Statistical inference is constructed on
information geometry, where the Fisher metric plays a fundamental role, whereas
Wasserstein geometry is useful for applications to computer vision and AI. We
propose statistical inference based on the Wasserstein geometry in the case
that the base space is 1-dimensional. By using the location-scale model, we
derive the $W$-estimator explicitly and studies its asymptotic behaviors.
