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

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 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. A general reparametrization method is suggested, which applies in all models where Fisher information misbehaves.

Based on joint work with Christophe Ley.

References
Azzalini, A. (1985). A class of distributions which includes the normal ones, Scandinavian
Journal of Statististics 12, 171–178.
Hallin, M. and Ley, Chr. (2012). Skew-symmetric distributions and Fisher information—a tale
of two densities, Bernoulli 18, 747–763.
Hallin, M. and Ley, Chr. (2014). Skew-symmetric distributions and Fisher information: the
double sin of the skew-normal, Bernoulli, to appear.