The complex Bingham quartic distribution and shape analysis

Abstract

The complex Bingham distribution was introduced by Kent as a tractable model for landmark-based shape analysis. It forms an exponential family with a sufficient statistic which is quadratic in the data. However, the distribution has too much symmetry to be widely useful. In particular, under high concentration it behaves asymptotically as a normal distribution, but where the covariance matrix is constrained to have complex symmetry. To overcome this limitation and to provide a full range of asymptotic normal behaviour, we introduce a new 'complex Bingham quartic distribution' by adding a selection of quartic terms to the log-density. In the simplest case this new distribution corresponds to Kent's FB5-distribution. Asymptotic and saddlepoint methods are developed for the normalizing constant to facilitate maximum likelihood estimation. Examples are given to show the usefulness of this new distribution. © 2006 Royal Statistical Society.