Skew-Symmetric Distributions and Fisher Information. The Double Sin of the Skew-Normal

Hallin and Ley (2012) investigate and fully characterize the Fisher singularity phenomenonin univariate and multivariate families of skew-symmetric distributions. Thispaper proposes a refined analysis of the (univariate) Fisher degeneracy problem, showingthat it can be more or less severe, inducing n1/4 (“simple singularity”), n1/6 (“doublesingularity”), 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 arepossible for generalized skew-normal families only. We also show that higher-ordersingularities, leading to worse-than-n1/8 rates, cannot occur.