| Summary: | 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.
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