An Efficient Test for Homogeneity of Mean Directions on the Hyper-sphere

The paper aims to develop a universally implementable efficient test for testing homogeneity of mean directions of several independent hyper-spherical populations. Conventional tests are valid only under highly concentrated and/or large-size groups. Focusing on the popular Langevin distribution on a d-hyper-sphere, the present work extends the very recent results for the circular case. The hurdle of the nuisance non-location-scale concentration parameter kappa is overcome through a variant of the integrated likelihood ratio test (ILRT), yielding a simple and elegant test statistic. Analytically, second-order accurate asymptotic chi-squared distribution of ILRT is established. Extensive simulation study demonstrates that ILRT uniformly outperforms its peers, notably under highly dispersed groups, which is precisely the target parametric region, and is robust under a large class of alternate distributions. Five real-life data analyses from diverse disciplines, including the emerging field of vectorcardiography and a novel application to compositional data analysis in the context of drug development, illustrate applications of the findings.