Refined saddlepoint approximations for the normalizing constant of the complex Bingham quartic distribution: A change of variable approach

Abstract

The complex Bingham quartic (CBQ) distribution is defined on the unit complex sphere in ℂ 𝑘−1 and it is relevant for the statistical shape analysis of a 𝑘-point landmark data in 2D. This extended the Fisher distribution on the unit spherical shape space 𝑆 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 approximations without a change of variable approach