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几何是引入概率分布流形中的两种主要结构，它们捕捉了分布流形的不同特征。我们基于Li与Zhao（2023）的框架，研究仿射变形统计模型中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几何的估计量的优缺点。在Wasserstein几何中，概率分布的形态及其仿射变形被分离，显示出其对波形扰动的鲁棒性，但以牺牲Fisher效率为代价。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。它仅当且仅当波形为高斯分布时，才与信息几何估计量（最大似然估计量）一致。Wasserstein效率在波形变化鲁棒性方面的作用得以阐明。