Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction of optimal tests for symmetry on the torus and their quantitative error bounds

Andreas Anastasiou, Christophe Ley, Sophia Loizidou

TL;DR

This paper develops optimal symmetry tests for data on the $d$-dimensional torus using Le Cam’s asymptotic experiment framework and Uniform Local Asymptotic Normality (ULAN). It introduces the sine-skewed toroidal model, proves a general quadratic mean differentiability result to support ULAN, and derives parametric and semi-parametric tests for both known and unknown symmetry centers. Finite-sample error bounds are obtained via Stein’s method, and extensive simulations plus a protein-folding data application illustrate robust performance and the practical need for skewed-toroidal modeling in real data. The work yields universally optimal tests against sine-skewed alternatives, provides explicit asymptotic distributions under null and local alternatives, and delivers a broad methodological toolkit for symmetry testing on the torus with quantitative error control.

Abstract

In this paper, we develop optimal tests for symmetry on the hyper-dimensional torus, leveraging Le Cam's methodology. We address both scenarios where the center of symmetry is known and where it is unknown. These tests are not only valid under a given parametric hypothesis but also under a very broad class of symmetric distributions. The asymptotic behavior of the proposed tests is studied both under the null hypothesis and local alternatives, and we derive quantitative bounds on the distributional distance between the exact (unknown) distribution of the test statistic and its asymptotic counterpart using Stein's method. The finite-sample performance of the tests is evaluated through simulation studies, and their practical utility is demonstrated via an application to protein folding data. Additionally, we establish a broadly applicable result on the quadratic mean differentiability of functions, a key property underpinning the use of Le Cam's approach.

Construction of optimal tests for symmetry on the torus and their quantitative error bounds

TL;DR

This paper develops optimal symmetry tests for data on the -dimensional torus using Le Cam’s asymptotic experiment framework and Uniform Local Asymptotic Normality (ULAN). It introduces the sine-skewed toroidal model, proves a general quadratic mean differentiability result to support ULAN, and derives parametric and semi-parametric tests for both known and unknown symmetry centers. Finite-sample error bounds are obtained via Stein’s method, and extensive simulations plus a protein-folding data application illustrate robust performance and the practical need for skewed-toroidal modeling in real data. The work yields universally optimal tests against sine-skewed alternatives, provides explicit asymptotic distributions under null and local alternatives, and delivers a broad methodological toolkit for symmetry testing on the torus with quantitative error control.

Abstract

In this paper, we develop optimal tests for symmetry on the hyper-dimensional torus, leveraging Le Cam's methodology. We address both scenarios where the center of symmetry is known and where it is unknown. These tests are not only valid under a given parametric hypothesis but also under a very broad class of symmetric distributions. The asymptotic behavior of the proposed tests is studied both under the null hypothesis and local alternatives, and we derive quantitative bounds on the distributional distance between the exact (unknown) distribution of the test statistic and its asymptotic counterpart using Stein's method. The finite-sample performance of the tests is evaluated through simulation studies, and their practical utility is demonstrated via an application to protein folding data. Additionally, we establish a broadly applicable result on the quadratic mean differentiability of functions, a key property underpinning the use of Le Cam's approach.

Paper Structure

This paper contains 22 sections, 12 theorems, 156 equations, 3 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Family of distributions and the ULAN property
  3. A technical result about Quadratic Mean Differentiability
  4. Family of sine-skewed distributions
  5. ULAN property
  6. Optimal tests for symmetry about a known center
  7. Optimal tests for unknown symmetry center
  8. Simulation results
  9. Application: protein data
  10. Conclusion
  11. Proofs
  12. Proof of Proposition \ref{['prop: QMD']}
  13. Proof of Proposition \ref{['prop: ULAN']}
  14. Proof of Theorem \ref{['thm: optimality specified median']}
  15. Proof of Theorem \ref{['thm: steins method specified']}
  16. ...and 7 more sections

Key Result

Proposition 1

Consider the functions $f(\boldsymbol{x};\boldsymbol{\mu}), g(\boldsymbol{x};\boldsymbol{\mu},\boldsymbol{\lambda})$, $f, g: \mathcal{X}\subset \mathbb{R}^m \rightarrow \mathbb{R}^+$ with parameters $\boldsymbol{\mu} \in \mathbb{R}^k, \boldsymbol{\lambda} \in \mathbb{R}^m$. Assume that $f(\boldsymbo Then $f(\boldsymbol{x}; \boldsymbol{\mu})$ is QMD with quadratic mean $\nabla_{\boldsymbol{\mu}} f^

Figures (3)

  • Figure 1: Results of Monte Carlo estimates of percentage of rejection of different null hypotheses of specified center symmetry, with increasing sample size and $f_0$ being the TWC with parameters $\rho_{12}=\rho_{23}=1, \rho_{13}=0.25$ and wrapped Cauchy marginal distributions with parameters $\beta_1=0.1, \beta_2=0.2, \beta_3=0.3$. The black line corresponds to the test $\mathcal{H}^{(n)}_{0; \mu;1}:\lambda_1=0 \text{ vs } \mathcal{H}^{(n)}_{1; \mu;1}:\lambda_1\neq0$, the red and green lines correspond to the analogous tests for $\lambda_2$ and $\lambda_3$ and the blue line corresponds to the test $\mathcal{H}^{(n)}_{0; \mu}:\boldsymbol{\lambda} = 0 \text{ vs } \mathcal{H}^{(n)}_{1; \mu}:\boldsymbol{\lambda} \neq 0$. The purple dotted line indicates 0.05.
  • Figure 2: Plots of the theoretical (light blue) and simulated (red) power of the test for (a) $\phi^{\ast (n); \mu}$ with $f_0 = BWC_{0.1, 0.5, 0.4}$, (b) $\phi^{\ast (n)}_{f_0}$ with $f_0 = g_0 = BWC_{0.9, 0.9, 0.4}$, (c) $\phi^{\ast (n)}_{f_0}$ with $f_0 = BWC_{0.8, 0.8, 0.6}$, $g_0 = BWC_{0.3, 0.3, 0.4}$ and (d) $\phi^{\ast (n)}_{f_0}$ with $f_0 = S_{0.2, 0.2, 0.4}$, $g_0 = S_{0.1, 0.1, 0.1}$.
  • Figure 3: Rose plots of the protein data. The theoretical mean $(-1.05, -0.87, \pi)$ is plotted in red and the estimated one $(-1.21, -0.33, 3.08)$ is plotted in blue.

Theorems & Definitions (24)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Proposition 3
  • Proposition 4
  • Theorem 3
  • Theorem 4
  • Lemma 2
  • ...and 14 more