Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants

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 χ12 random variabl...

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Bibliographic Details
Published in:Biometrika 2005-06, Vol.92 (2), p.465-476
Main Authors: Kume, A., Wood, Andrew T. A.
Format: Article
Language:English
Subjects:
Applications
Approximation
Biology, psychology, social sciences
Complex Bingham distribution
Density
Directional data
Distribution theory
Eigenvalues
Exact sciences and technology
Mathematical constants
Mathematical vectors
Mathematics
Maximum likelihood estimation
Maximum likelihood estimators
Modified Bessel functions
Multivariate analysis
Probability and statistics
Probability theory and stochastic processes
Random variables
Sciences and techniques of general use
Shape analysis
Statistical analysis
Statistics
Unit vectors
Von Mises-Fisher distribution
Watson distribution
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Description
Summary: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 χ12 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.
ISSN:0006-3444
1464-3510
DOI:10.1093/biomet/92.2.465