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

Eduardo García-Portugués

TL;DR

The paper introduces the spherical cardioid distribution $\mathrm{C}_{k}(\boldsymbol\mu,\rho)$ on $\mathbb{S}^{d}$ as a tractable, rotationally symmetric model that generalizes the circular cardioid. It develops a full inference framework including method-of-moments and maximum-likelihood estimators with explicit asymptotic results, and a bootstrap-based projected-ecdf goodness-of-fit test that leverages closed-form projected distributions. A comprehensive set of numerical experiments demonstrates estimator performance and GOF behavior, while an application to long-period comet orbits shows the model’s practical utility in capturing mild non-uniformity with axial symmetry. The work also establishes convolution-closure properties and exact moment/characteristic-function expressions, facilitating connections to related directional distributions and enabling robust statistical testing on the sphere.

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.

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

TL;DR

The paper introduces the spherical cardioid distribution on as a tractable, rotationally symmetric model that generalizes the circular cardioid. It develops a full inference framework including method-of-moments and maximum-likelihood estimators with explicit asymptotic results, and a bootstrap-based projected-ecdf goodness-of-fit test that leverages closed-form projected distributions. A comprehensive set of numerical experiments demonstrates estimator performance and GOF behavior, while an application to long-period comet orbits shows the model’s practical utility in capturing mild non-uniformity with axial symmetry. The work also establishes convolution-closure properties and exact moment/characteristic-function expressions, facilitating connections to related directional distributions and enabling robust statistical testing on the sphere.

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.
Paper Structure (31 sections, 15 theorems, 220 equations, 7 figures, 3 tables, 3 algorithms)

This paper contains 31 sections, 15 theorems, 220 equations, 7 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background on orthogonal polynomials
  3. Spherical cardioid distribution
  4. Genesis
  5. Closedness under convolution
  6. Moments
  7. Characteristic function
  8. Simulation
  9. Estimation and inference
  10. Method of moments
  11. Maximum likelihood
  12. Asymptotic relative efficiency
  13. Goodness-of-fit
  14. Projected distributions
  15. Test statistic
  16. ...and 16 more sections

Key Result

Proposition 3.1

Let $\boldsymbol\mu_1,\boldsymbol\mu_2\in\mathbb{S}^{d}$, $\rho_1,\rho_2\in[-1,1]$, and $k_1,k_2\geq1$. Then, and, if $\boldsymbol{X}\mid \boldsymbol\Xi\sim \mathrm{C}_{k_1}(\boldsymbol\Xi,\rho_1)$ and $\boldsymbol\Xi\sim \mathrm{C}_{k_2}(\boldsymbol\mu_2,\rho_2)$, then $\boldsymbol{X}\sim \mathrm{C}_{k_1}(\boldsymbol\mu_2,\delta_{k_1,k_2} \rho_1\rho_2/d_{k_1,d})$.

Figures (7)

  • Figure 1: Samples and density of the spherical cardioid on $\mathbb{S}^1$ with $\boldsymbol\mu=\boldsymbol{e}_2$ (red point, north) and $(k,\rho)\in\{(1,1), (2, 1), (2, -1), (3, 1), (4, 1), (4, -1)\}$. For each panel, a random sample of $n=200$ observations is shown. The dashed curve gives the uniform density $1/(2\pi)$ as reference. Negative-$\rho$ panels illustrate overparametrization.
  • Figure 2: Samples and density of the spherical cardioid on $\mathbb{S}^2$ with $\boldsymbol\mu=\boldsymbol{e}_3$ (red point, north) and $(k,\rho)\in\{(1,1), (2, 1), (2, -1), (3, 1), (4, 1), (4, -1)\}$. The plots show the front hemisphere of $\mathbb{S}^2$, with shading applied to the points in the back hemisphere. The sample, with $n=2000$ observations, is colored according to the value of the density at the observations.
  • Figure 3: Asymptotic relative efficiencies $\rho\mapsto\mathrm{ARE}_{\mathrm{MM}}(\boldsymbol\mu)$ and $\rho\mapsto\mathrm{ARE}_{\mathrm{MM}}(\rho)$ for $k=1,2$ and $d=1,\ldots,10$.
  • Figure 4: Asymptotic relative efficiencies $\rho\mapsto\mathrm{ARE}_{\mathrm{GM}}(\rho)$ for $k=3$ and $d=1,\ldots,10$ (left panel) and $k=1,\ldots,6$ and $d=2$ (right panel).
  • Figure 5: Histograms of $\{\sqrt{n}(\hat{\mu}^{(j)}_1-\mu_1)\}_{j=1}^M$ and $\{\sqrt{n}(\hat{\rho}^{(j)}-\rho)\}_{j=1}^M$ against their asymptotic normal density, for $k=1,2$ and $d=2$. Inside each panel, the left plot corresponds to the maximum likelihood estimator and the right plot corresponds to the method of moments estimator. The gray/red dashed lines indicate the empirical/asymptotic means.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Definition 3.1: Spherical cardioid distribution
  • Proposition 3.1: Closedness under convolution
  • Theorem 3.1: Vectorized moments
  • Corollary 3.1: Specific moments
  • Corollary 3.2: Covariance matrices of vectorized moments
  • Proposition 3.2: Characteristic and moment generating functions
  • Proposition 3.3
  • Theorem 4.1: Moment estimators when $k=1$
  • Theorem 4.2: Moment estimators when $k=2$
  • Theorem 4.3: Estimation of $\rho$ via Gegenbauer moments
  • ...and 21 more