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 the waveform is Gaussian. The role of the Wasserstein efficiency is elucidated in terms of robustness against waveform change.

翻译：信息几何与Wasserstein几何是概率分布流形中引入的两种主要结构，它们刻画了流形的不同特性。我们在Li与Zhao（2023）的框架下研究仿射变形统计模型中的Wasserstein几何特征，该模型是位置-尺度模型的多维推广。我们比较了基于信息几何与Wasserstein几何的估计量的优缺点。在Wasserstein几何中，概率分布的形态与其仿射变形被分离开来，这显示出其在以损失Fisher效率为代价的条件下对波形扰动的鲁棒性。我们证明在椭圆对称仿射变形模型的情形下，Wasserstein估计量即为矩估计量。当波形为高斯分布时，该估计量与信息几何估计量（最大似然估计量）一致。我们通过其对波形变化的鲁棒性阐明了Wasserstein效率的作用。