A Cram\'er-von Mises test of uniformity on the hypersphere

TL;DR

This paper introduces a projection-based Cramér-von Mises test for assessing uniformity on the hypersphere, extending the Watson test, with practical null distribution and an application to crater distribution on Venus.

Contribution

It presents a new statistical test for uniformity on the hypersphere, including its asymptotic distribution and practical implementation.

Findings

Null distribution is tractable and practical.

Test successfully applied to crater distribution on Venus.

Extension of Watson test to higher-dimensional spheres.

Abstract

Testing uniformity of a sample supported on the hypersphere is one of the first steps when analysing multivariate data for which only the directions (and not the magnitudes) are of interest. In this work, a projection-based Cram\'er-von Mises test of uniformity on the hypersphere is introduced. This test can be regarded as an extension of the well-known Watson test of circular uniformity to the hypersphere. The null asymptotic distribution of the test statistic is obtained and, via numerical experiments, shown to be tractable and practical. A novel study on the uniformity of the distribution of craters on Venus illustrates the usage of the test.