## Bernoulli

• Bernoulli
• Volume 20, Number 3 (2014), 1432-1453.

### Skew-symmetric distributions and Fisher information: The double sin of the skew-normal

#### 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.

#### Article information

Source
Bernoulli, Volume 20, Number 3 (2014), 1432-1453.

Dates
First available in Project Euclid: 11 June 2014

https://projecteuclid.org/euclid.bj/1402488945

Digital Object Identifier
doi:10.3150/13-BEJ528

Mathematical Reviews number (MathSciNet)
MR3217449

Zentralblatt MATH identifier
1302.60030

#### Citation

Hallin, Marc; Ley, Christophe. Skew-symmetric distributions and Fisher information: The double sin of the skew-normal. Bernoulli 20 (2014), no. 3, 1432--1453. doi:10.3150/13-BEJ528. https://projecteuclid.org/euclid.bj/1402488945

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