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**Saddlepoint approximations for the normalizing constant of Fisher-Bingham distributions on products of spheres and Stiefel manifolds.**
*(English)*
Zbl 1452.62998

Summary: In an earlier paper the first and third author [ibid. 92, No. 2, 465–476 (2005; Zbl 1094.62063)] 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.

### MSC:

62R30 | Statistics on manifolds |

62E17 | Approximations to statistical distributions (nonasymptotic) |

62H11 | Directional data; spatial statistics |