Rotation-Coordinate Density Estimation on Spheres

Abstract

We study a coordinate-based kernel density estimator for spherical directional data. The construction represents each point of the sphere by a selected rotation and performs smoothing with a distance inherited from the rotation group. The method is formulated as a section-based procedure, not as a fully intrinsic rotation-invariant estimator. We derive local bias, variance, mean squared error and consistency properties away from the polar coordinate singularities. To address numerical instability near the poles, we add a locally normalized pole-corrected version inspired by boundary correction techniques. Simulations show that this correction substantially reduces polar errors.