Rate-optimal and computationally efficient nonparametric estimation on the circle and the sphere

TL;DR

This paper introduces new density estimators on the circle and sphere that are both rate-optimal and computationally efficient, with applications across various scientific fields.

Contribution

The authors develop novel density estimators that achieve optimal rates and are computationally practical for data on the circle and sphere.

Findings

Derived closed-form probability estimates for regions on the circle and sphere.

Supported theories with extensive simulation studies.

Demonstrated applicability in diverse scientific case studies.

Abstract

We investigate the problem of density estimation on the unit circle and the unit sphere from a computational perspective. Our primary goal is to develop new density estimators that are both rate-optimal and computationally efficient for direct implementation. After establishing these estimators, we derive closed-form expressions for probability estimates over regions of the circle and the sphere. Then, the proposed theories are supported by extensive simulation studies. The considered settings naturally arise when analyzing phenomena on the Earth's surface or in the sky (sphere), as well as directional or periodic phenomena (circle). The proposed approaches are broadly applicable, and we illustrate their usefulness through case studies in zoology, climatology, geophysics, and astronomy, which may be of independent interest. The methodologies developed here can be readily applied across a wide range of scientific domains.