Skew-Symmetric Distributions and Fisher Information the double sin of the skew-normal

Hallin and Ley (2012) 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) Fisher degeneracy problem, showing that it 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.