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

Abstract

In this paper, we study the spherical cardioid distribution, a higher-dimensional and arbitrary-order generalization of the circular cardioid distribution. This distribution is rotationally symmetric and generates unimodal, multimodal, axial, and girdle-like densities. We identify various properties 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 of order $k$ up to order $k-1$ 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 bootstrap goodness-of-fit tests based on the projected empirical cumulative distribution function 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.

26 pages, 5 figures, 3 tables. Supplementary material: 34 pages, 4 figures, 4 tables