  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.
