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.

翻译：信息几何与Wasserstein几何是概率分布流形上引入的两种主要结构，它们捕捉了该流形的不同特性。我们在Li与Zhao（2023）提出的仿射变形统计模型框架下研究Wasserstein几何的特性，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几何的估计量的优劣。概率分布的形状及其仿射变形在Wasserstein几何中得以分离，这表明该估计量在牺牲Fisher效率的同时对波形扰动具有鲁棒性。我们证明，在椭圆对称仿射变形模型情形下，Wasserstein估计量即为矩估计量。当且仅当波形为高斯分布时，该估计量与信息几何估计量（极大似然估计量）一致。本文阐明了Wasserstein效率在应对波形变化鲁棒性方面的作用。