On the spherical cardioid distribution and its goodness-of-fit

TL;DR

This paper introduces the spherical cardioid distribution, detailing its properties, estimation methods, and a bootstrap goodness-of-fit test, demonstrating its applicability to real-world data such as comet orbits.

Contribution

It provides a comprehensive study of the spherical cardioid distribution, including its characteristics, estimators, and a new goodness-of-fit testing framework.

Findings

Distribution is rotationally symmetric and generates various density shapes.

Moments match those of the uniform distribution on the sphere.

Application to comet orbit data demonstrates practical usefulness.

Abstract

In this paper, we study the spherical cardioid distribution, a higher-dimensional and higher-order generalization of the circular cardioid distribution. This distribution is rotationally symmetric and generates unimodal, multimodal, axial, and girdle-like densities. We show several characteristics of the spherical cardioid that make it highly tractable: simple density evaluation, closedness under convolution, explicit expressions for vectorized moments, and efficient simulation. The moments of the spherical cardioid up to a given order coincide with those of the uniform distribution on the sphere, highlighting its closeness to the latter. We derive estimators by the method of moments and maximum likelihood, their asymptotic distributions, and their asymptotic relative efficiencies. We give the machinery for a bootstrap goodness-of-fit test based on the projected-ecdf approach, including the projected distribution and closed-form expressions for test statistics. An application to modeling the orbits of long-period comets shows the usefulness of the spherical cardioid distribution in real data analyses.