Harmonic analysis and distributionfree 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 modelfree inference such as testing uniformity and symmetry etc. in dealing with 2dimensional directions. However a similar discussion for spherical distributions, which are used to model 3dimensional 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 largesample tests for these symmetries, are provided. The paper concludes with a realdata example analyzing the longitudinal sunspot activity.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 arXiv:
 arXiv:1710.00253
 Bibcode:
 2017arXiv171000253R
 Keywords:

 Statistics  Methodology;
 62H11;
 62H15
 EPrint:
 26 pages, 2 figures