Fisher Geometry of Circular Wrapped Normals: Completion and Curvature
Abstract
We determine the intrinsic Fisher geometry of the two-parameter circular wrapped normal family. Although its finite parameters form a cylinder, its metric completion is a disk: adjoining the uniform law extends the Fisher metric to a positive-definite real-analytic metric. The Gaussian curvature vanishes at the added point, is positive nearby, and tends to −1/2 as the underlying standard deviation tends to zero. We prove that the curvature has exactly one nonuniform zero radius and exactly one critical radius, the latter a nondegenerate positive maximum. Opposite points sufficiently close to uniformity are joined by a unique minimizing geodesic through the uniform law, so their intrinsic distance is not attained inside the nonuniform family. In contrast, identifying the means of real normal distributions modulo a period retains a complete hyperbolic cylinder with an infinitely distant cusp. Wrapping strictly decreases Fisher information in every nonzero tangent direction, while the information deficit and its fixed-order derivatives are exponentially small as the standard deviation tends to zero. The curvature theorem is proved by convergent series and explicit analytic remainder estimates; numerical figures serve only as illustrations.
Related
- Submanifolds of higher rank with curvature normals of constant length
- The Wrapped Q-Lindley Distribution: A New Circular Model with Statistical Properties, Classical and Bayesian Inference, and Applications to Directional Data
- Constant mean curvature disks with circular boundary
- Constant mean curvature disks with circular boundary in the $3$-sphere and in hyperbolic space
- Stable constant mean curvature disks with circular boundary in homogeneous three-manifolds
BibTeX
@misc{jang2026fisher,
title = {{Fisher Geometry of Circular Wrapped Normals: Completion and Curvature}},
author = {Hyun Chul Jang and Juna Lee and Nathan Negera and Kalea Plotnick and Antony Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37985},
eprint = {2610.37985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37985},
}
RIS
TY - UNPB
TI - Fisher Geometry of Circular Wrapped Normals: Completion and Curvature
AU - Hyun Chul Jang
AU - Juna Lee
AU - Nathan Negera
AU - Kalea Plotnick
AU - Antony Wang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37985
UR - https://arxiv.org/abs/2610.37985
AB - We determine the intrinsic Fisher geometry of the two-parameter circular wrapped normal family. Although its finite parameters form a cylinder, its metric completion is a disk: adjoining the uniform law extends the Fisher metric to a positive-definite real-analytic metric. The Gaussian curvature vanishes at the added point, is positive nearby, and tends to −1/2 as the underlying standard deviation tends to zero. We prove that the curvature has exactly one nonuniform zero radius and exactly one critical radius, the latter a nondegenerate positive maximum. Opposite points sufficiently close to uniformity are joined by a unique minimizing geodesic through the uniform law, so their intrinsic distance is not attained inside the nonuniform family. In contrast, identifying the means of real normal distributions modulo a period retains a complete hyperbolic cylinder with an infinitely distant cusp. Wrapping strictly decreases Fisher information in every nonzero tangent direction, while the information deficit and its fixed-order derivatives are exponentially small as the standard deviation tends to zero. The curvature theorem is proved by convergent series and explicit analytic remainder estimates; numerical figures serve only as illustrations.
ER -