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Skew-symmetric distributions and Fisher information: The double sin of the skew-normal
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Skew-symmetric distributions and Fisher information: The double sin of the skew-normal
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Journal Article
Skew-symmetric distributions and Fisher information: The double sin of the skew-normal
2014
Overview
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\") n⅙ (\"double singularity\"), or n1/8 (\"triple singularity\") consistency rates for the skewness parameter. We show, however, that simple singularity (yielding n⅙ 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-n⅛ 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-f families, by Azzalini [Scand. J. Stat. 12 (1985) 171-178], Arei lano-Val le and Azzalini [J. Multivariate Anal. 113 (2013) 73-90], and DiCiccio and Monti [Quaderni di Statistica 13 (2011) 1-21], respectively.
Publisher
International Statistical Institute and Bernoulli Society for Mathematical Statistics and Probability,Bernoulli Society for Mathematical Statistics and Probability
Subject
