# Flexible Class of Skew‐Symmetric Distributions

Abstract: We propose a flexible class of skew-symmetric distributions for which the probab- ility density function has the form of a product of a symmetric density and a skewing function. By constructing an enumerable dense subset of skewing functions on a compact set, we are able to consider a family of distributions, which can capture skewness, heavy tails and multimodality systematically. We present three illustrative examples for the fibreglass data, the simulated data from a mixture of two normal distributions and the Swiss bills dlata.

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621 citations

### Cites background from "Flexible Class of Skew‐Symmetric Di..."

...This important problem has been examined byWang et al. (2004) and Ma & Genton (2004), whose results can be summarized as follows; in their formulation, the term Gfw(x)g in (12) is replaced by the perturbation function p(x), satisfying conditions (14)....

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### Cites background from "Flexible Class of Skew‐Symmetric Di..."

...One of these extensions refers to the family of flexible skew-symmetric (FSS) distributions introduced by Ma & Genton (2004) with density fFSS(x ; ξ, ω, α1, . . . , αK ) = 2ω−1 f0(z) G{PK (z)}, x ∈ R, (5) where f0 and G are symmetric univariate density function and distribution function,…...

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...Other forms of skew-t distribution have been considered by Jones & Faddy (2003), Sahu et al. (2003) and Ma & Genton (2004)....

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182 citations

### Cites background or methods from "Flexible Class of Skew‐Symmetric Di..."

...(24) Under regularity conditions on the pdffV, Ma & Genton (2004) have proved that the class of FSSdistributions is dense in the class ofSSdistributions under theL∞ norm....

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...Flexible skewed distributions have been introduced by Ma & Genton (2004)....

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##### References

2,470 citations

### "Flexible Class of Skew‐Symmetric Di..." refers background or methods in this paper

...This representation has been used by Azzalini & Capitanio (2003) to define certain distributions by perturbation of symmetry....

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...For K = 1, the pdf is always unimodal as was already noted by Azzalini (1985) for the univariate skew-normal distribution....

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...The case K = 1 corresponds to Azzalini & Dalla Valle's (1996) bivariate skew-normal distribution, which cannot capture the bimodality....

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...For K ¼ 1, the pdf is always unimodal as was already noted by Azzalini (1985) for the univariate skew-normal distribution....

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...In particular, for ,il = ,B2 = /33 = & = 0, the pdf is exactly the bivariate skew-normal proposed by Azzalini & Dalla Valle (1996), and known to be unimodal (see Fig....

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### "Flexible Class of Skew‐Symmetric Di..." refers background in this paper

...Finally, note that the stochastic representation of FSSdistributions follows from the stochastic representation of SS distributions described byWang et al. (2004), see also Azzalini & Capitanio (2003)....

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...Similarly, multivariate distributions such as skew-t (Branco & Dey, 2001; Azzalini & Capitanio, 2003; Jones & Faddy, 2003; Sahu et al., 2003), skew-Cauchy (Arnold & Beaver, 2000) and other skewelliptical ones (Azzalini & Capitanio, 1999; Branco & Dey, 2001; Sahu et al., 2003) can be represented by…...

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...Jones & Faddy (2003) and Azzalini & Capitanio (2003) fit two forms of skew-t distributions to these data....

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...If each term has an odd order (all ks are odd), then the polynomial is called an odd polynomial, whereas if each term has an even order (all ks are even), it is called an even polynomial....

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