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On the singularity of the Fisher Information matrix in the sine-skewed family on the d-dimensional torus

Emily Schutte, Sophia Loizidou, Vincent Laheurte

Abstract

Skewed distributions are fundamental in modelling asymmetric data on the d-dimensional torus. In this context, asymmetry is introduced through the sine-skewing mechanism, which is the only skewing mechanism that has been proposed on the hyper-torus in the literature. Some sine-skewed models are known to suffer from a singular Fisher information matrix in the vicinity of symmetry, which poses a significant issue for inferential purposes. It is an open question to determine for which sine-skewed models Fisher information singularity occurs. In this paper, a general characterization of the class of models that exhibit this singularity is given in the general d-dimensional setting.

On the singularity of the Fisher Information matrix in the sine-skewed family on the d-dimensional torus

Abstract

Skewed distributions are fundamental in modelling asymmetric data on the d-dimensional torus. In this context, asymmetry is introduced through the sine-skewing mechanism, which is the only skewing mechanism that has been proposed on the hyper-torus in the literature. Some sine-skewed models are known to suffer from a singular Fisher information matrix in the vicinity of symmetry, which poses a significant issue for inferential purposes. It is an open question to determine for which sine-skewed models Fisher information singularity occurs. In this paper, a general characterization of the class of models that exhibit this singularity is given in the general d-dimensional setting.
Paper Structure (5 sections, 1 theorem, 20 equations)

This paper contains 5 sections, 1 theorem, 20 equations.

Table of Contents

  1. Introduction
  2. Fisher information singularity for the $d$-dimensional torus
  3. Examples of known distributions
  4. Discussion
  5. Proof of Theorem 1

Key Result

Theorem 1

Let $f_0\in\mathcal{F}$ satisfy Assumption assumption. The FIM of the sine-skewed version of $f_0(\boldsymbol{\theta} {- \boldsymbol{\mu}})$ in the vicinity of symmetry is singular if and only if there exists a vector $\boldsymbol{\alpha} = \left( \alpha_1, \ldots, \alpha_d \right)^\intercal\in\math for all $t\in \mathbb{R}$ and $\boldsymbol{\theta} \in [-\pi,\pi)^d$.

Theorems & Definitions (2)

  • Theorem 1
  • proof