Flatness of location-scale-shape models under the Wasserstein metric
Ayumu Fukushi, Yoshinori Nakanishi-Ohno, Takeru Matsuda
TL;DR
This work extends location-scale models by introducing the location-scale-shape family with a shape parameter $\xi$ that controls tail heaviness, enabling application to extreme-value theory within the Wasserstein geometric framework. The authors derive the Wasserstein score functions and the full Wasserstein information matrix in terms of the moment-generating function $M_f$, showing that the model is intrinsically flat yet not extrinsically flat under the $L^2$-Wasserstein metric, due to its warped-product structure. They provide explicit geometric constructions: an intrinsic flatness proof via a coordinate transformation to a unit-metric frame, and a demonstration that displacement interpolations do not, in general, preserve the family, except in the special case $\xi_1=\xi_2$. These results advance geometric statistics in the Wasserstein space and offer a foundation for Wasserstein-based inference in heavy-tailed models such as the GEV and GPD, with future work planned for multivariate extensions and statistical properties.
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.