Refined saddlepoint approximations for the normalizing constant of the complex Bingham quart

Abstract

The complex Bingham quartic (CBQ) distribution is defined on the unit complex sphere in          and it is relevant for the statistical shape analysis of a    -point landmark data in 2D. This extended the Fisher distribution on the unit spherical shape space           2 (1/2). The complex Bingham quartic (CBQ) distribution provides suitable shape parameters to comprise anisotropy. Under high concentrations, it looks like a multivariate Gaussian normal distribution but the main drawback of this planar shape distribution is that its normalizing constant does not have a simple closed explicit form representation. The present paper provides a modified approximation procedure for the indeterminate normalizing constant of the CBQ distribution based on saddlepoint approximations with a change of variable scheme. The modified saddlepoint approximations under a change of variable seem more precise as compared with the saddlepoint approximation without a change of variable approach.