The epsilon–skew–normal distribution for analyzing near-normal data

Mudholkar, Govind S.; Hutson, Alan D.

doi:10.1016/s0378-3758(99)00096-8

Cited by 201 publications

(138 citation statements)

References 23 publications

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“…In the two-piece normal case, I m = 1, I h = 2/π and I s = 2. The asymptotic variance-covariance matrix of the ML estimators of µ, γ and σ then corresponds with that given in Theorem 4.7 of Mudholkar & Hutson (2000) (except that they give it for σ 2 rather than σ). Mudholkar & Hutson go on to note that the asymptotic correlation betweenμ andγ is 4/ √ 6π ≈ 0.921, which is the value associated with δ → ∞ in Section 3.3.…”

Section: Example: the Two-piece (Skew) Normal Distributionmentioning

confidence: 92%

“…For concreteness we take k = 2 and set a(γ) = 1 − h(γ), b(γ) = 1 + h(γ) where −1 < h(γ) < 1. The simplest choice h(γ) = γ was made by Mudholkar & Hutson (2000) and preferred in inferential work by Arellano-Valle et al (2005) and Cassart et al (2008); it is an effective one. However, asymptotic independence between skewness and the scale and shape/tailweight parameters is sensitive to the skewness parametrisation used.…”

Section: Likelihood Fitting In the Arellano-valle Et Al Parametrisationmentioning

confidence: 99%

“…When f = φ, the standard normal density, density (3.1) without δ reduces to the two-piece skew-normal distribution (dating to Fechner, 1897) in the parametrisation suggested by Mudholkar & Hutson (2000) (who use ǫ in place of γ and set a(ǫ) = 1 − ǫ). This is also, of course, the limiting case of the two-piece t distribution as δ → ∞.…”

Section: Example: the Two-piece (Skew) Normal Distributionmentioning

confidence: 99%

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On parameter orthogonality in symmetric and skew models

Jones

Anaya-Izquierdo

2011

Journal of Statistical Planning and Inference

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Orthogonal and partly orthogonal reparametrisations are provided for certain wide and important families of univariate continuous distributions. First, the orthogonality of parameters in location-scale symmetric families is extended to symmetric distributions involving a third parameter. This sets the scene for consideration of the four-parameter situation in which skewness is also allowed. It turns out that one specific approach to generating such fourparameter families, that of two-piece distributions with a certain parametrisation restriction, has some attractive features with regard to parameter orthogonality. This work also affords partly orthogonal parametrisations of three-parameter two-piece models.

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Section: Example: the Two-piece (Skew) Normal Distributionmentioning

confidence: 92%

Section: Likelihood Fitting In the Arellano-valle Et Al Parametrisationmentioning

confidence: 99%

Section: Example: the Two-piece (Skew) Normal Distributionmentioning

confidence: 99%

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On parameter orthogonality in symmetric and skew models

Jones

Anaya-Izquierdo

2011

Journal of Statistical Planning and Inference

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“…1(b), are true 'skewnormal' distributions in the sense of admitting normality as well as asymmetry and, by (6), retaining two normal-like tails. They share this property with 'two-piece' normal distributions (Fechner, 1897, Fernandez and Steel, 1998, Mudholkar and Hutson, 2000 in which two differentially scaled halves of a normal distribution are joined together. However, unlike the two-piece normal, density (2) is infinitely differentiable at all x ∈ R. The current and two-piece densities differ from the popular skew-normal distribution with density 2φ(x)Φ(λx) (Azzalini, 1985, Genton, 2004 for which a side-effect of introducing the skewness parameter λ is a change to the weight in one of the tails.…”

Section: An Asymmetric Subfamilymentioning

confidence: 99%

Sinh-arcsinh distributions

Jones¹,

Pewsey²

2009

Biometrika

231

182

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Summary. We introduce the 'sinh-arcsinh transformation' and thence, by applying it to random variables from some 'generating' distribution with no further parameters beyond location and scale (which we take for most of the paper to be the normal), a new family of 'sinh-arcsinh distributions'. This four parameter family has both symmetric and skewed members and allows for tailweights that are both heavier and lighter than those of the generating distribution. The 'central' place of the normal distribution in this family affords likelihood ratio tests of normality that appear to be superior to the state-of-the-art because of the range of alternatives against which they are very powerful. Likelihood ratio tests of symmetry are also available and very successful. Three-parameter symmetric and asymmetric subfamilies of the full family are of interest too. Heavy-tailed symmetric sinh-arcsinh distributions behave like Johnson S U distributions while light-tailed symmetric sinh-arcsinh distributions behave like Rieck and Nedelman's sinh-normal distributions, the sinh-arcsinh family allowing a seamless transition between the two, via the normal, controlled by a single parameter. The sinh-arcsinh family is very tractable and many properties are explored. Likelihood inference is pursued, including an attractive reparametrisation. A multivariate version is considered. Options and extensions are discussed.

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“…are normalizing functions, and δ ∈ R again plays the role of a skewness parameter. This appealing and easily interpretable skewing scheme has been used by several authors, such as, e.g., Fechner (1897), Fernàndez and Steel (1998), or Mudholkar and Hutson (2000), which defined the so-called epsilon-skew-normal distributions. Of particular interest is the inverse scale factors model from Fernàndez and Steel (1998), where a 2 (δ) = δ = (b 2 (δ)) −1 .…”

Section: Introductionmentioning

confidence: 99%

Multivariate skewing mechanisms: A unified perspective based on the transformation approach

Ley

Paindaveine

2010

Statistics & Probability Letters

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In recent years, models for (possibly multivariate) skewed distributions have become more and more popular. In the univariate case, Ferreira and Steel (2006) [Ferreira, J.T.A.S., Steel, M.F.J., 2006. A constructive representation of univariate skewed distributions. J. Amer. Statist. Assoc. 101, 823-829] introduced general skewing mechanisms in order to compare existing skewing methods in a common framework and to ease construction of new such methods according to the needs in given situations. In this paper, we make use of the classical transformation approach to define alternative skewing mechanisms for the same purpose. While keeping all nice features of Ferreira and Steel's skewing mechanisms (flexibility, surjectivity, possibility of retaining prespecified characteristics of the original symmetric distribution, etc.), our skewing mechanisms, unlike theirs, can easily be extended to the multivariate case. We describe our skewing schemes, investigate their main properties, and illustrate their effects on standard (multi)normal distributions by means of a few examples. Finally, we briefly discuss their relevance in the context of optimal symmetry testing.

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The epsilon–skew–normal distribution for analyzing near-normal data

Cited by 201 publications

References 23 publications

On parameter orthogonality in symmetric and skew models

On parameter orthogonality in symmetric and skew models

Sinh-arcsinh distributions

Multivariate skewing mechanisms: A unified perspective based on the transformation approach

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