Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powers

Eduardo García-Portugués, Davy Paindaveine, Thomas Verdebout

Abstract

We consider a class of symmetry hypothesis testing problems including testing isotropy on $\mathbb{R}^d$ and testing rotational symmetry on the hypersphere $\mathcal{S}^{d-1}$. For this class, we study the null and non-null behaviors of Sobolev tests, with emphasis on their consistency rates. Our main results show that: (i) Sobolev tests exhibit a detection threshold (see Bhattacharya, 2019, 2020) that does not only depend on the coefficients defining these tests; and (ii) tests with non-zero coefficients at odd (respectively, even) ranks only are blind to alternatives with angular functions whose $k$th-order derivatives at zero vanish for any $k$ odd (even). Our non-standard asymptotic results are illustrated with Monte Carlo exercises. A case study in astronomy applies the testing toolbox to evaluate the symmetry of orbits of long- and short-period comets.

On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powers

Abstract

We consider a class of symmetry hypothesis testing problems including testing isotropy on and testing rotational symmetry on the hypersphere . For this class, we study the null and non-null behaviors of Sobolev tests, with emphasis on their consistency rates. Our main results show that: (i) Sobolev tests exhibit a detection threshold (see Bhattacharya, 2019, 2020) that does not only depend on the coefficients defining these tests; and (ii) tests with non-zero coefficients at odd (respectively, even) ranks only are blind to alternatives with angular functions whose th-order derivatives at zero vanish for any odd (even). Our non-standard asymptotic results are illustrated with Monte Carlo exercises. A case study in astronomy applies the testing toolbox to evaluate the symmetry of orbits of long- and short-period comets.

Paper Structure

This paper contains 12 sections, 14 theorems, 115 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Testing problems and the proposed class of Sobolev tests
  3. Spherical harmonics, Sobolev tests, and behavior under the null hypothesis
  4. Asymptotics under the alternative: non-null moments of random spherical harmonics
  5. Asymptotics under the alternative: local non-null behavior of Sobolev tests
  6. Finite Sobolev tests
  7. Infinite Sobolev tests
  8. Numerical illustrations
  9. Symmetry of comet orbits
  10. Perspectives for future research
  11. Proofs of Section \ref{['sec:moments']}
  12. Proofs of Section \ref{['sec:localpowfiniteinf']}

Key Result

Proposition 1

Fix an integer $p \geq 3$ and a positive integer $k$. Let $\mathcal{M}_k=\{{\bf m} \in\mathbb{N}_0^p: |{\bf m}|:=m_1+\cdots+m_p=k \textrm{ and }m_p\in\{0,1\}\}$. For any ${\bf m}\in\mathcal{M}_k$, let where we write $|{\bf m}^{j}|=m_j+\cdots+m_{p}$ and $\lambda_j:=|{\bf m}^{j+1}|+(p-j-1)/2$ for any $j=1,\ldots,p-2$, and where $b_{\bf m}:=2$ if $m_{p-1}+m_p>0$ and $b_{\bf m}:=1$ otherwise. Then,

Figures (4)

  • Figure 1: Rejection frequencies of the Rayleigh test (green), Bingham test (blue), and $3$-test (dark orange), all conducted at asymptotic level $\alpha=5\%$ for samples generated by the rotationally symmetric model with angular function $f(s)=\exp(s)$ and concentration $\kappa_n=n^{-1/\ell}\tau$, for $\ell=2,4,6,12$ and $\tau\in\{0,0.5,1,\ldots,6\}$. Whenever the rejection frequencies are non-trivial (that is, strictly between $\alpha$ and $1$), the corresponding asymptotic powers are also provided in solid lines.
  • Figure 2: Same experiment as in Figure \ref{['Fig1']}, but with samples generated from a rotationally symmetric model with angular function $f(s)=\exp(s^2)$ and $\tau\in\{0,0.5,1,\ldots,4\}$.
  • Figure 3: Same experiment as in Figure \ref{['Fig1']}, but with samples generated from a rotationally symmetric model with angular function $f(s)=\exp(s^3)$ and $\tau\in\{0,0.5,1,\ldots,3\}$.
  • Figure 4: Orbits of long- and short-period comets and their normal vectors. Within each figure, the left plot displays ten illustrative elliptical orbits and their associated normal vectors, with the ecliptic plane shown in gray and the Sun represented as an orange sphere (one focus of the elliptical orbits). The right plot in each figure shows the full dataset of normal orbit vectors on $\mathcal{S}^2$.

Theorems & Definitions (14)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • ...and 4 more