Skew-symmetric distributions and Fisher information: The double sin of the skew-normal

TL;DR

This paper refines the understanding of Fisher singularities in skew-symmetric distributions, revealing that simple singularities are most common and higher-order singularities are rare, with implications for parameter estimation.

Contribution

It provides a detailed analysis of the severity of Fisher singularities in skew-normal families and proposes a reparametrization to address these issues.

Findings

Simple singularity is the most common case.

Double and triple singularities are only possible in generalized skew-normal families.

Higher-order singularities cannot occur, ensuring a lower bound on estimation difficulty.

Abstract

Hallin and Ley [Bernoulli 18 (2012) 747-763] investigate and fully characterize the Fisher singularity phenomenon in univariate and multivariate families of skew-symmetric distributions. This paper proposes a refined analysis of the (univariate) problem, showing that singularity can be more or less severe, inducing $n^{1/4}$ ("simple singularity"), $n^{1/6}$ ("double singularity"), or $n^{1/8}$ ("triple singularity") consistency rates for the skewness parameter. We show, however, that simple singularity (yielding $n^{1/4}$ consistency rates), if any singularity at all, is the rule, in the sense that double and triple singularities are possible for generalized skew-normal families only. We also show that higher-order singularities, leading to worse-than-$n^{1/8}$ rates, cannot occur. Depending on the degree of the singularity, our analysis also suggests a simple reparametrization that offers an alternative to the so-called centred parametrization proposed, in the particular case of skew-normal and skew-$t$ families, by Azzalini [Scand. J. Stat. 12 (1985) 171-178], Arellano-Valle and Azzalini [J. Multivariate Anal. 113 (2013) 73-90], and DiCiccio and Monti [Quaderni di Statistica 13 (2011) 1-21], respectively.