论文标题
一维定位尺度模型中的Wasserstein统计数据
Wasserstein Statistics in One-dimensional Location-Scale Model
论文作者
论文摘要
Wasserstein几何形状和信息几何形状是在概率分布的多种形式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来定义的，因此它反映了定义分布的基本歧管的度量。信息几何形状定义为在基本空间的可逆转换下是不变的。两者都有自己的应用程序。尤其是，统计推断是基于信息几何形状，其中Fisher指标起着基本作用，而Wasserstein几何形状在计算机视觉和AI应用中很有用。在这项研究中，我们分析了基于Wasserstein几何形状的统计推断，即基本空间是一维的。通过使用位置尺度模型，我们进一步得出了W-静态器，该W-估计器明确地将运输成本从经验分布到统计模型并研究其渐近行为。我们表明，W-估计器是一致的，并且通过使用功能增量方法明确地给出了其渐近分布。在高斯情况下，W-估计器在高斯案件中具有效率。
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 particular, statistical inference is based upon information geometry, where the Fisher metric plays a fundamental role, whereas Wasserstein geometry is useful in computer vision and AI 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.
