Wasserstein geometry and information geometry are two important structures to be introduced in a manifold of probability distributions. Wasserstein geometry is defined by using the transportation cost between two distributions, so it reflects the metric of the base manifold on which the distributions are defined. Information geometry is defined to be invariant under reversible transformations of the base space. Both have their own merits for applications. In this study, we analyze statistical inference based on the Wasserstein geometry in the case that the base space is one-dimensional. By using the location-scale model, we further derive the W-estimator that explicitly minimizes the transportation cost from the empirical distribution to a statistical model and study its asymptotic behaviors. We show that the W-estimator is consistent and explicitly give its asymptotic distribution by using the functional delta method. The W-estimator is Fisher efficient in the Gaussian case.

Wasserstein几何学和信息几何学是在概率分布的多样性中引入的两个重要结构。Wasserstein几何形状是通过使用两个分布之间的运输成本定义的，因此它反映了定义分布的基础歧管的度量。信息几何形状被定义为在基本空间的可逆转换下是不变的。两者在应用方面都有各自的优点。在本研究中，在基础空间为一维的情况下，我们基于Wasserstein几何分析统计推断。通过使用位置比例模型，我们进一步推导了W估计量，该量显着最小化了从经验分布到统计模型的运输成本，并研究了其渐近行为。我们表明W-估计量是一致的，并通过使用函数德尔塔方法显式给出其渐近分布。在高斯情况下，W估计器是费雪有效的。