A new distance-based distribution: Detecting concentration in directional data

Abstract

In this article, we propose a simple method to study high concentrations in spherical data, in particular in the directional perspective. To this end, we define a distance-based distribution in the interval $(0,1)$, called the T-statistic distance ($\mathit{DT}$) law, to describe the scattering of points on the unit sphere. Our model is derived from the von Mises–Fisher (vMF) distribution, which is one of the most known directional laws. We show that if the data are vMF distributed, their concentration can be modeled by our distribution. Some of its properties are derived and discussed: Moment generating function, kurtosis and skewness. Likelihood-based inference procedures are provided for both points and hypotheses concerning the $\mathit{DT}$ concentration. Further, we propose a new test statistic in terms of the $\mathit{DT}$ distribution to deal with scattering within spherical data and derive its exact density. Numerical studies show that maximum likelihood estimates behave asymptotically well even for small sample sizes and that the likelihood ratio test for the $\mathit{DT}$ distribution often performs better than Wald tests. We apply our model to paleomagnetic data to illustrate how it is used to analyze spherical data concentration. Results show that the distance-based approach works well to identify high concentration on the unit sphere.