Skew-Symmetric distributions and fisher information: The double sin of the skew-Normal

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 n1/4 ("simple singularity"), n 1/6("double singularity"), or n1/8 ("triple singularity") consistency rates for the skewness parameter.We show, however, that simple singularity (yielding n1/4 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-N1/8 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. ©2014 ISI/BS.