Saddlepoint approximations for the normalizing constant of Fisher–Bingham distributions on products of spheres and Stiefel manifolds
In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher– Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation. In this sequel, we extend the approach to a more general setting and derive s...
| Main Authors: | , , |
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| Format: | Article |
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Oxford University Press
2013
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| Online Access: | https://eprints.nottingham.ac.uk/2446/ |
| _version_ | 1848790787626631168 |
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| author | Kume, A. Preston, S.P. Wood, Andrew T.A. |
| author_facet | Kume, A. Preston, S.P. Wood, Andrew T.A. |
| author_sort | Kume, A. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher–
Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint
density approximation. In this sequel, we extend the approach to a more general setting
and derive saddlepoint approximations for the normalizing constants of multicomponent Fisher–
Bingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on
Stiefel manifolds. In each case, the approximation for the normalizing constant is essentially
a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic
forms in normal variables. Both first-order and second-order saddlepoint approximations are considered.
Computational algorithms, numerical results and theoretical properties of the approximations
are presented. In the challenging high-dimensional settings considered in this paper the
saddlepoint approximations perform very well in all examples considered.
Some key words: Directional data; Fisher matrix distribution; Kent distribution; Orientation statistics. |
| first_indexed | 2025-11-14T18:18:10Z |
| format | Article |
| id | nottingham-2446 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:18:10Z |
| publishDate | 2013 |
| publisher | Oxford University Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-24462020-05-04T16:38:29Z https://eprints.nottingham.ac.uk/2446/ Saddlepoint approximations for the normalizing constant of Fisher–Bingham distributions on products of spheres and Stiefel manifolds Kume, A. Preston, S.P. Wood, Andrew T.A. In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher– Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation. In this sequel, we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent Fisher– Bingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on Stiefel manifolds. In each case, the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables. Both first-order and second-order saddlepoint approximations are considered. Computational algorithms, numerical results and theoretical properties of the approximations are presented. In the challenging high-dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered. Some key words: Directional data; Fisher matrix distribution; Kent distribution; Orientation statistics. Oxford University Press 2013-08-13 Article PeerReviewed Kume, A., Preston, S.P. and Wood, Andrew T.A. (2013) Saddlepoint approximations for the normalizing constant of Fisher–Bingham distributions on products of spheres and Stiefel manifolds. Biometrika, 100 (4). pp. 971-984. ISSN 0006-3444 http://biomet.oxfordjournals.org/content/100/4/971.full doi:10.1093/biomet/ast021 doi:10.1093/biomet/ast021 |
| spellingShingle | Kume, A. Preston, S.P. Wood, Andrew T.A. Saddlepoint approximations for the normalizing constant of Fisher–Bingham distributions on products of spheres and Stiefel manifolds |
| title | Saddlepoint approximations for the normalizing constant of
Fisher–Bingham distributions on products of spheres and
Stiefel manifolds |
| title_full | Saddlepoint approximations for the normalizing constant of
Fisher–Bingham distributions on products of spheres and
Stiefel manifolds |
| title_fullStr | Saddlepoint approximations for the normalizing constant of
Fisher–Bingham distributions on products of spheres and
Stiefel manifolds |
| title_full_unstemmed | Saddlepoint approximations for the normalizing constant of
Fisher–Bingham distributions on products of spheres and
Stiefel manifolds |
| title_short | Saddlepoint approximations for the normalizing constant of
Fisher–Bingham distributions on products of spheres and
Stiefel manifolds |
| title_sort | saddlepoint approximations for the normalizing constant of
fisher–bingham distributions on products of spheres and
stiefel manifolds |
| url | https://eprints.nottingham.ac.uk/2446/ https://eprints.nottingham.ac.uk/2446/ https://eprints.nottingham.ac.uk/2446/ |