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Measuring deviations from spherical symmetry

Lujia Bai, Holger Dette

TL;DR

This work introduces a principled measure of deviation from spherical symmetry for a p-dimensional vector by quantifying the distance between the joint density of $(\|Y\|, Y/\|Y\|)$ and its best spherical approximation, yielding an explicit $L^2$-distance form $\mathcal{M}^2$. It develops a kernel-based, $U$-statistic estimator $\hat{\mathcal{M}}^2_n$ with asymptotic normality that depends on whether spherical symmetry holds, and provides both asymptotic and pivotal inference tools (including self-normalized inference) for constructing confidence intervals and testing relevant deviations with a threshold $\Delta$. The paper also delivers finite-sample results via simulations, bias-reduction strategies, and a real-data example on exchange rates, illustrating practical procedures for assessing approximate, rather than exact, sphericity. Overall, the methodology enables robust quantification and inference about deviations from spherical symmetry with applicability across multivariate settings and dependent data contexts.

Abstract

Most of the work on checking spherical symmetry assumptions on the distribution of the $p$-dimensional random vector $Y$ has its focus on statistical tests for the null hypothesis of exact spherical symmetry. In this paper, we take a different point of view and propose a measure for the deviation from spherical symmetry, which is based on the minimum distance between the distribution of the vector $\big (\|Y\|, Y/ \|Y\| )^\top $ and its best approximation by a distribution of a vector $\big (\|Y_s\|, Y_s/ \|Y_s \| )^\top $ corresponding to a random vector $Y_s$ with a spherical distribution. We develop estimators for the minimum distance with corresponding statistical guarantees (provided by asymptotic theory) and demonstrate the applicability of our approach by means of a simulation study and a real data example.

Measuring deviations from spherical symmetry

TL;DR

This work introduces a principled measure of deviation from spherical symmetry for a p-dimensional vector by quantifying the distance between the joint density of and its best spherical approximation, yielding an explicit -distance form . It develops a kernel-based, -statistic estimator with asymptotic normality that depends on whether spherical symmetry holds, and provides both asymptotic and pivotal inference tools (including self-normalized inference) for constructing confidence intervals and testing relevant deviations with a threshold . The paper also delivers finite-sample results via simulations, bias-reduction strategies, and a real-data example on exchange rates, illustrating practical procedures for assessing approximate, rather than exact, sphericity. Overall, the methodology enables robust quantification and inference about deviations from spherical symmetry with applicability across multivariate settings and dependent data contexts.

Abstract

Most of the work on checking spherical symmetry assumptions on the distribution of the -dimensional random vector has its focus on statistical tests for the null hypothesis of exact spherical symmetry. In this paper, we take a different point of view and propose a measure for the deviation from spherical symmetry, which is based on the minimum distance between the distribution of the vector and its best approximation by a distribution of a vector corresponding to a random vector with a spherical distribution. We develop estimators for the minimum distance with corresponding statistical guarantees (provided by asymptotic theory) and demonstrate the applicability of our approach by means of a simulation study and a real data example.
Paper Structure (18 sections, 12 theorems, 135 equations, 1 figure, 5 tables)

This paper contains 18 sections, 12 theorems, 135 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. A measure of deviation and its estimation
  3. Asymptotic properties and first statistical applications
  4. Asymptotic theory
  5. Some statistical applications
  6. Pivotal inference
  7. Finite sample properties
  8. Confidence intervals for the deviation from a spherical distribution
  9. Relevant hypothesis
  10. Data analysis
  11. Preliminary technical results
  12. Proof of main results
  13. Proof of \ref{['mindist']}
  14. Martingale structure under spherical symmetry
  15. Proof of \ref{['prop:meanvar']}
  16. ...and 3 more sections

Key Result

Proposition 2.1

The minimum in det103 is obtained for the marginal density of $(U,V) = ( \|Y\|, Y/\|Y\|)$, that is $h^*(u):= f_U(u) = \int_{\mathbb{S}^{p-1}} f(u,v) \ \omega_{p-1}(\mathrm dv)$, and given by

Figures (1)

  • Figure 1: Absolute autocorrelations of the log-returns of the daily exchange rate of the Yen to the Dollar and the Pound to the Euro from January 2nd, 2009, to December 31st, 2009.

Theorems & Definitions (19)

  • Proposition 2.1
  • Proposition 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Corollary 3.1
  • Corollary 3.2
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.1
  • ...and 9 more