High-order parameter approximation for von Mises–Fisher distributions

This paper concerns the issue of the maximum-likelihood estimation (MLE) for the concentration parameters of the von Mises–Fisher (vMF) distributions, which are crucial to directional data analysis. In particular, we study the numerical approximation approach for solving the implicit nonlinear equation arising from building the MLE of the concentration parameter κκ of vMF distributions. In addition, we address the implementation of Is(x)Is(x), the modified Bessel function of the first kind, which is the most time-consuming and fundamental ingredient in the proposed approximation scheme of the MLE for κκ. The main contribution of this paper is two fold. The first is to present a two-steps Halley based method for exploring a high-order approximation of the MLE for κκ, which can significantly contribute to the improvement of estimation accuracy. The second is to develop a novel approach for the implementation of Is(x)Is(x), which can make the substantial improvement of computation efficiency for computing the MLE approximation for κκ. The numerical experiments were conducted to compare the proposed schemes with those in the existing works by Tanabe et al. [1] and Sra [2]. The experimental results show that, given the same amount of computation as their methods, the proposed high-order scheme can achieve much more accurate approximations while our implementation of Is(x)Is(x) is preferable yet desirable for high dimensional applications.