Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants

A. Kume and Andrew T. A. Wood

Biometrika, 2005, vol. 92, issue 2, 465-476

Abstract: The Fisher--Bingham distribution is obtained when a multivariate normal random vector is conditioned to have unit length. Its normalising constant can be expressed as an elementary function multiplied by the density, evaluated at 1, of a linear combination of independent noncentral χ-sub-1-super-2 random variables. Hence we may approximate the normalising constant by applying a saddlepoint approximation to this density. Three such approximations, implementation of each of which is straightforward, are investigated: the first-order saddlepoint density approximation, the second-order saddlepoint density approximation and a variant of the second-order approximation which has proved slightly more accurate than the other two. The numerical and theoretical results we present showthat this approach provides highly accurate approximations in a broad spectrum of cases. Copyright 2005, Oxford University Press.

Date: 2005
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