The complex Bingham quartic distribution and shape analysis

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. Copyright 2006 Royal Statistical Society.