ABSTRACT
In this chapter, we consider two fundamental types of test procedures on hyperspheres, namely tests for uniformity and for symmetry. The problem of testing uniformity on the unit hypersphere S p - 1 $ \mathcal{S}^{p - 1} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119472/5ff6ea01-2f47-4320-8bbe-e544021cff79/content/inline-math6_1.tif"/> is one of the oldest problems in directional statistics. It can be traced back to the discussion by Bernoulli (1735) on the solution to the problem of whether the closeness of the orbital planes of various planets arose by chance or not. Rayleigh (1880) was the first to study the resultant length of bivariate uniform unit vectors, and the first test of uniformity was proposed in Rayleigh (1919). Since then, the problem has attracted considerable attention. Kuiper (1960) studied Kolmogorov-Smirnov type tests for the circular case while Watson (1961) introduced Cramér-von Mises type tests. Ajne (1968) tested circular uniformity by comparing the number of observations in each semicircle with the expected value of n/2; this test was extended to S p-1 by Beran (1968). That same paper, together with Beran (1969), introduced a class of circular tests that are locally most powerful invariant against a specific non-parametric alternative, while Giné (1975) proposed Sobolev tests of uniformity on the unit hypersphere. Bingham (1974) constructed tests of uniformity of axial data by exploiting the idea that the sample scatter matrix should, under uniformity, be close to I p /p. Cordeiro &Ferrari (1991) modified the Rayleigh test in the circular case in order to improve the chi-squared approximation at the limit as n → ∞. Jupp (2001) extended the results of Cordeiro & Ferrari (1991) to the hypersphere. We refer the reader to Sections 6.3, 8.3, 10.4.1, 10.7.1 and 10.8 of Mardia & Jupp (2000) for detailed information about most of the aforementioned tests.
