Title: Harmonic analysis and distribution-free inference for spherical distributions,
Abstract: 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 etc. in dealing with
2-dimensional directions. However a similar discussion for spherical
distributions, which are used to model 3-dimensional directional data, has not
been fully developed in the literature in terms of their harmonics. This paper,
in what we believe is the first such attempt, looks at the probability
distributions on a unit sphere, through the perspective of spherical harmonics,
analogous to the Fourier analysis for distributions on a unit circle. Harmonic
representations of many currently used spherical models are presented and
discussed. A very general family of spherical distributions is then introduced,
special cases of which yield many known spherical models. Through the prism of
harmonic analysis, one can look at the mean direction, dispersion, and various
forms of symmetry for these models in a generic setting. Aspects of
distribution free inference such as estimation and large-sample tests for these
symmetries, are provided. The paper concludes with a real-data example
analyzing the longitudinal sunspot activity.