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<p>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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月12日

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  • Ayumu Fukushi,
  • Yoshinori Nakanishi-Ohno,
  • Takeru Matsuda

摘要

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.