Flatness of location-scale-shape models under the Wasserstein metric

2025/11/13 by Fukushi, Ayumu, Nakanishi-Ohno, Yoshinori, Matsuda, Takeru
#62F99 (Primary) 53B20 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2511.09959

Abstract

In Wasserstein geometry, one-dimensional location-scale models are flat both intrinsically and extrinsically-that is, they are curvature-free as well as totally geodesic in the space of probability distributions. In this study, we introduce a class of one-dimensional statistical models, termed the location-scale-shape model, which generalizes several distributions used in extreme-value theory. This model has a shape parameter that specifies the tail heaviness. We investigate the Wasserstein geometry of the location-scale-shape model and show that it is intrinsically flat but extrinsically curved.

Citations

• On the attainment of the Wasserstein--Cramer--Rao lower bound
• Information Geometry of Wasserstein Statistics on Shapes and Affine Deformations
• On a prior based on the Wasserstein information matrix
• Wasserstein Statistics in One-dimensional Location-Scale Model
• Wasserstein information matrix
• Ricci curvature for parametric statistics via optimal transport
• Optimal transport natural gradient for statistical manifolds with continuous sample space

Cited by

• Wasserstein projection estimators for circular distributions
• Refining Cramér-Rao Bound With Multivariate Parameters: An Extrinsic Geometry Perspective
• Improving Cramér-Rao Bound And Its Variants: An Extrinsic Geometry Perspective

Related

• Wasserstein statistics in 1D location-scale model
• Wasserstein Statistics in One-dimensional Location-Scale Model
• Warped metrics for location-scale models
• Information Geometry of Wasserstein Statistics on Shapes and Affine Deformations
• Statistical data analysis in the Wasserstein space