Saddlepoint approximations for the Bingham and Fisher-Bingham normalising constants

• 成果类型: Article
• 署名作者: Kume, A; Wood, ATA
• 署名单位: University of Kent; University of Nottingham
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/92.2.465
• 发表日期: 2005
• 页码: 465476
• 关键词: sphere; statistics

摘要

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(2)(1) 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 show that this approach provides highly accurate approximations in a broad spectrum of cases.