Uniform asymptotics and entropy structure of the Bingham distribution in arbitrary dimensions
Abstract
The Bingham distribution is widely used to model directional data with antipodal symmetry, and its normalizing constant Z and moments play a central role in statistical inference and closure models. Their behavior becomes singular when one or more eigenvalue gaps of the parameter matrix become unbounded. In this work, we develop a uniform asymptotic framework for the Bingham distribution in arbitrary dimensions. Starting from an inverse-Laplace integral representation, we derive asymptotic expansions with explicit remainder estimates that are uniform with respect to arbitrary relative scales among multiple diverging eigenvalues. In a particular regime, the asymptotic series becomes absolutely convergent with an exponentially small remainder. These results yield precise asymptotic profiles of the Bingham moments and characterize the degeneration of higher-dimensional distributions to lower-dimensional counterparts. As an application to the Bingham closure, we prove that the entropy admits a decomposition into an explicit logarithmic leading term and a residual that is uniformly Lipschitz continuous on the moment simplex. Moreover, on each boundary face, the residual agrees with its lower-dimensional version up to an explicit additive constant. The results provide a unified description of the singular structure of the Bingham distribution and have implications for numerical closure and Q-tensor models of nematic liquid crystals.
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BibTeX
@misc{wu2026uniform,
title = {{Uniform asymptotics and entropy structure of the Bingham distribution in arbitrary dimensions}},
author = {Dawei Wu and Lei Zhang and Pingwen Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07121},
eprint = {2609.07121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07121},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Uniform asymptotics and entropy structure of the Bingham distribution in arbitrary dimensions
AU - Dawei Wu
AU - Lei Zhang
AU - Pingwen Zhang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.07121
UR - https://arxiv.org/abs/2609.07121
AB - The Bingham distribution is widely used to model directional data with antipodal symmetry, and its normalizing constant Z and moments play a central role in statistical inference and closure models. Their behavior becomes singular when one or more eigenvalue gaps of the parameter matrix become unbounded. In this work, we develop a uniform asymptotic framework for the Bingham distribution in arbitrary dimensions. Starting from an inverse-Laplace integral representation, we derive asymptotic expansions with explicit remainder estimates that are uniform with respect to arbitrary relative scales among multiple diverging eigenvalues. In a particular regime, the asymptotic series becomes absolutely convergent with an exponentially small remainder. These results yield precise asymptotic profiles of the Bingham moments and characterize the degeneration of higher-dimensional distributions to lower-dimensional counterparts. As an application to the Bingham closure, we prove that the entropy admits a decomposition into an explicit logarithmic leading term and a residual that is uniformly Lipschitz continuous on the moment simplex. Moreover, on each boundary face, the residual agrees with its lower-dimensional version up to an explicit additive constant. The results provide a unified description of the singular structure of the Bingham distribution and have implications for numerical closure and Q-tensor models of nematic liquid crystals.
ER -