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The complex Bingham quartic distribution and shape analysis
Author(s) -
Kent J. T.,
Mardia K. V.,
McDonnell P.
Publication year - 2006
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/j.1467-9868.2006.00565.x
Subject(s) - quartic function , exponential family , mathematics , distribution (mathematics) , asymptotic distribution , quadratic equation , exponential distribution , symmetry (geometry) , statistical physics , mathematical analysis , pure mathematics , statistics , physics , geometry , estimator
Summary. 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 FB 5 ‐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.