Harmonic analysis and distribution-free inference for spherical distributions

Fourier analysis, and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry, in dealing with 2-dimensional directions. However, a similar discussion for spherical distributions, which are used to model 3-dimensional directional data, is not readily available in the literature in terms of their harmonics. This paper, in what we believe is the first such attempt, looks at probability distributions on a unit sphere through the perspective of spherical harmonics, analogous to the Fourier analysis for distributions on a unit circle. Representation of any continuous spherical density in terms of spherical harmonics is given, and such series expansions provided for some commonly used spherical distributions, as well as for two new spherical distributions that are introduced. Through the prism of harmonic analysis, one can look at the mean direction, dispersion, and various forms of symmetry for these models in a nonparametric setting. Aspects of distribution-free inference such as estimation and large-sample tests for various symmetries, are provided, each type of symmetry being characterized through its harmonics. The paper concludes with a real-data example analyzing the longitudinal sunspot activity.