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

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Families of skew-symmetric distributions (Azzalini 1985) are subject to Fisher information singularity problems under symmetry. We investigate this phenomenon, showing that it can be more or less severe, inducing n1/4 (“simple singularity”), n1/6 (“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. A general reparametrization method is suggested, which applies in all models where Fisher information misbehaves.

Based on joint work with Christophe Ley

This talk is part of the Statistics series.

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