Skew-symmetric distributions and Fisher information -- a tale of two densities

TL;DR

This paper investigates the Fisher information singularity in skew-symmetric distributions, revealing the conditions under which it occurs and clarifying the role of Gaussian densities in this phenomenon.

Contribution

It provides a comprehensive characterization of information singularity in multivariate skew-symmetric densities and clarifies the relationship between symmetric kernels and skewing functions.

Findings

Characterization of Fisher information singularity conditions

Link between symmetric kernels and skewing functions

Gaussian densities' role in singularity phenomenon

Abstract

Skew-symmetric densities recently received much attention in the literature, giving rise to increasingly general families of univariate and multivariate skewed densities. Most of those families, however, suffer from the inferential drawback of a potentially singular Fisher information in the vicinity of symmetry. All existing results indicate that Gaussian densities (possibly after restriction to some linear subspace) play a special and somewhat intriguing role in that context. We dispel that widespread opinion by providing a full characterization, in a general multivariate context, of the information singularity phenomenon, highlighting its relation to a possible link between symmetric kernels and skewing functions -- a link that can be interpreted as the mismatch of two densities.