Information geometry and Wasserstein geometry are two main structures
introduced in a manifold of probability distributions, and they capture its
different characteristics. We study characteristics of Wasserstein geometry in
the framework of Li and Zhao (2023) for the affine deformation statistical
model, which is a multi-dimensional generalization of the location-scale model.
We compare merits and demerits of estimators based on information geometry and
Wasserstein geometry. The shape of a probability distribution and its affine
deformation are separated in the Wasserstein geometry, showing its robustness
against the waveform perturbation in exchange for the loss in Fisher
efficiency. We show that the Wasserstein estimator is the moment estimator in
the case of the elliptically symmetric affine deformation model. It coincides
with the information-geometrical estimator (maximum-likelihood estimator) when
and only when the waveform is Gaussian. The role of the Wasserstein efficiency
is elucidated in terms of robustness against waveform change