Saddlepoint approximations for the normalizing constant of Fisher–Bingham distributions on products of spheres and Stiefel manifolds

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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

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Abstract

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.

Item Type: Article
RIS ID: https://nottingham-repository.worktribe.com/output/717114
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: https://doi.org/10.1093/biomet/ast021
Depositing User: de Sousa, Mrs Shona
Date Deposited: 15 Apr 2014 10:09
Last Modified: 04 May 2020 16:38
URI: https://eprints.nottingham.ac.uk/id/eprint/2446

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