Skew-symmetric distributions and Fisher information

The double sin of the skew-normal

M. Hallin, C. Ley

    Research output: Contribution to journalArticleScientificpeer-review

    Abstract

    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 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. 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-t families, by Azzalini [Scand. J. Stat. 12 (1985) 171–178], Arellano-Valle and Azzalini [J. Multivariate Anal. 113 (2013) 73–90], and DiCiccio and Monti [Quaderni di Statistica 13 (2011) 1–21], respectively.
    Original languageEnglish
    Pages (from-to)1432-1453
    JournalBernoulli
    Volume20
    Issue number3
    DOIs
    Publication statusPublished - 2014

    Fingerprint

    Symmetric Distributions
    Fisher Information
    Skew
    Singularity
    Univariate
    Normal Family
    Singularity Analysis
    Reparametrization
    Skewness
    Bernoulli
    Parametrization
    Higher Order
    Alternatives

    Cite this

    Hallin, M. ; Ley, C. / Skew-symmetric distributions and Fisher information : The double sin of the skew-normal. In: Bernoulli. 2014 ; Vol. 20, No. 3. pp. 1432-1453.
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    abstract = "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 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. 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-t families, by Azzalini [Scand. J. Stat. 12 (1985) 171–178], Arellano-Valle and Azzalini [J. Multivariate Anal. 113 (2013) 73–90], and DiCiccio and Monti [Quaderni di Statistica 13 (2011) 1–21], respectively.",
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    Skew-symmetric distributions and Fisher information : The double sin of the skew-normal. / Hallin, M.; Ley, C.

    In: Bernoulli, Vol. 20, No. 3, 2014, p. 1432-1453.

    Research output: Contribution to journalArticleScientificpeer-review

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    T1 - Skew-symmetric distributions and Fisher information

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    AU - Ley, C.

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    AB - 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 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. 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-t families, by Azzalini [Scand. J. Stat. 12 (1985) 171–178], Arellano-Valle and Azzalini [J. Multivariate Anal. 113 (2013) 73–90], and DiCiccio and Monti [Quaderni di Statistica 13 (2011) 1–21], respectively.

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