Estimation of the Bingham distribution function on nearly two-dimensional data sets

Summary. In cases where directional data, such as palaeomagnetic directions, lie nearly along a great circle, a good approximation to the maximum likelihood estimate of the intermediate concentration parameter k ₂ in the Bingham probability distribution is given by: 2( t ₂/ N ) – 1 = I ₁(1/2 k ₂)/ I ₀(1/2 k ₂), where t ₂ is the intermediate eigenvalue, N is the number of samples, and the I_i are the appropriate modified Bessel functions of the first kind. This estimate, the asymptotic limit as the smallest eigenvalue t ₁→ 0, corresponds to restricting all points to lie on a great circle. The limit is also useful as an endpoint for interpolation, especially since numerical calculation in this region is difficult.