  While spherical data arises in many contexts, including in directional
statistics, the current tools for density estimation and population comparison
on spheres are quite limited. Popular approaches for comparing populations (on
Euclidean domains) mostly involvea two-step procedure: (1) estimate probability
density functions (pdfs) from their respective samples, most commonly using the
kernel density estimator, and, (2) compare pdfs using a metric such as the L2
norm. However, both the estimated pdfs and their differences depend heavily on
the chosen kernels, bandwidths, and sample sizes. Here we develop a framework
for comparing spherical populations that is robust to these choices.
Essentially, we characterize pdfs on spherical domains by quantifying their
smoothness. Our framework uses a spectral representation, with densities
represented by their coefficients with respect to the eigenfunctions of the
Laplacian operator on a sphere. The change in smoothness, akin to using
different kernel bandwidths, is controlled by exponential decays in coefficient
values. Then we derive a proper distance for comparing pdf coefficients while
equalizing smoothness levels, negating influences of sample size and bandwidth.
This signifies a fair and meaningful comparisons of populations, despite vastly
different sample sizes, and leads to a robust and improved performance. We
demonstrate this framework using examples of variables on S1 and S2, and
evaluate its performance using a number of simulations and real data
experiments.
