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Biometrika 2005 92(2):465-476; doi:10.1093/biomet/92.2.465
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© 2005 Biometrika Trust

Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants

A. Kume1 and Andrew T. A. Wood2

1 Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT 27NF, U.K. a.kume{at}kent.ac.uk, 2 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, U.K. atw{at}maths.nott.ac.uk

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 {chi}12 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.

Key Words: Complex Bingham distribution; Directional data; Shape analysis; Von Mises-Fisher distribution; Watson distribution

Received June 2003. Revised November 2004.


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