Modern Perspectives on Hyperspherical Uniformity Testing: Maximal Projections and Stein Characterizations

We introduce a novel methodology for testing uniformity on the unit hypersphere $S^{d−1}$, based on maximal projections and recent developments in Stein’s method. The first framework provides a unified perspective that encompasses classical tests such as those of Rayleigh and Bingham, while also revealing connections to multivariate skewness and kurtosis. We derive the asymptotic null distribution using limit theorems for Banach space-valued stochastic processes and employ tools from spherical harmonics theory to simulate the corresponding limiting distributions. The test’s performance is analyzed under both contiguous and fixed alternatives, and consistency is established for a broad class of alternatives. Furthermore, we present Bahadur efficiency results for specific alternatives. Theoretical properties and empirical power are assessed through comprehensive Monte Carlo simulations. Complementing this, we explore a second, more recent ap- proach leveraging Stein characterizations to propose new testing procedures that extend the insights of the projection-based framework.

Keywords. uniformity tests, maximal projections, directional data, stochastic pro- cesses in Banach spaces, contiguous alternatives, Monte Carlo simulations, Stein’s method