Abstract: Tests for uniformity of distribution for data vectors on the d-dimensional hypersphere are proposed. The tests are U-statistic and V-statistic estimates of the quadratic distance between the hypothesized, under the null, uniform distribution on the sphere and the empirical cumulative distribution function. We introduce a class of diffusion kernels and study in detail a special member of this class, the Poisson kernel, on which our proposed tests of uniformity are based. We obtain the Karhunen-Loéve decomposition of the kernel, connect it with its degrees of freedom, and hence with the power of the test via a tuning parameter, the diffusion parameter. We propose an algorithm that allows one to select the tuning parameter, and study the connection between the Poisson kernel-based tests and the Sobolev tests. We then study the performance of the proposed tests in terms of level and power, for a number of alternative distributions. Our simulations show that the proposed methods are powerful and outperform the Rayleigh, Giné, Ajne and Bingham test procedures in the case of multimodal alternatives. We apply the new methods to test uniformity of data on the orbits of comets obtained from the NASA website.

Key words and phrases: Diffusion kernels, directional data, exit on the sphere distribution, multimodal alternatives, Poisson kernel, spherical data, testing for uniformity.