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

TL;DR

This paper explores the Wasserstein geometric properties of a new class of location-scale-shape models, revealing their intrinsic flatness but extrinsic curvature, and generalizing distributions in extreme-value theory.

Contribution

It introduces the location-scale-shape model and analyzes its Wasserstein geometric structure, extending understanding of flatness and curvature in probability distribution spaces.

Findings

The model is intrinsically flat in Wasserstein space.

The model exhibits extrinsic curvature in Wasserstein space.

Generalizes distributions used in extreme-value theory.

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.