Skewed multivariate models related to hidden truncation and/or selective reporting

Arnold, Barry C.; Beaver, Robert J.; Azzalini, Adelchi; Balakrishnan, N.; Bhaumik, A.; Dey, Dipak K.; Cuadras, Carles M.; Sarabia, José Marı́a

doi:10.1007/bf02595728

Cited by 221 publications

(136 citation statements)

References 16 publications

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“…The density f ADV(Σ,α),d given in (3) coincides with (2.4) of Arnold & Beaver (2002), which was proposed as a generalisation of the skew-Normal distribution of Azzalini (1985). As the measures of univariate skewness that we employ are invariant to location and scale transformation, we have that Sk m [ADV(Σ, α), d] is equal to the skewness of the distribution with density…”

Section: The Adv-normal Classmentioning

confidence: 78%

“…It is not clear at all how Sk(G y1 ) would relate to F , especially for high dimensional distributions. See Arnold & Beaver (2002) for a general discussion of conditional modelling and its advantages for interpretation. The practical disadvantage is computational.…”

Section: Directional Skewnessmentioning

confidence: 99%

“…For the distribution generated by (5) the moment generating function is given by (2.5) of Arnold & Beaver (2002). The maximum achievable skewness as measured by CE is the same as for the univariate skew-Normal of Azzalini (1985), and, thus, CE ∈ (−0.99527, 0.99527), where the bounds correspond to the values for the half-Normal.…”

Section: The Adv-normal Classmentioning

confidence: 99%

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On describing multivariate skewed distributions: A directional approach

Ferreira

Steel

2006

Can J Statistics

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Most multivariate measures of skewness in the literature measure the overall skewness of a distribution. These measures were designed for testing the hypothesis of distributional symmetry and their relevance for describing skewed distributions is less obvious. In this article, we consider the problem of characterising the skewness of multivariate distributions. We define directional skewness as the skewness along a direction and analyse two parametric classes of skewed distributions using measures based on directional skewness. The analysis brings further insight into the classes, allowing for a more informed selection of classes of distributions for particular applications. In the context of Bayesian linear regression under skewed error we use the concept of directional skewness twice. First in the elicitation of a prior on the parameters of the error distribution, and then in the analysis of the skewness of the posterior distribution of the regression residuals. Title in French:Résumé : Most multivariate measures of skewness in the literature measure the overall skewness of a distribution. These measures were designed for testing the hypothesis of distributional symmetry and their relevance for describing skewed distributions is less obvious. In this article, we consider the problem of characterising the skewness of multivariate distributions. We define directional skewness as the skewness along a direction and analyse parametric classes of skewed distributions using measures based on directional skewness. The analysis brings further insight into the classes, allowing for a more informed selection of particular classes for particular applications. In the context of Bayesian linear regression under skewed error we use the concept of directional skewness twice. First in the elicitation of a prior on the parameters of the error distribution, and then in the analysis of the skewness of the posterior distribution of the regression residuals.

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Section: The Adv-normal Classmentioning

confidence: 78%

Section: Directional Skewnessmentioning

confidence: 99%

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On describing multivariate skewed distributions: A directional approach

Ferreira

Steel

2006

Can J Statistics

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“…A random variable X λ is said to follow the skewed normal distribution SN(λ) if the probability density function (PDF) of X λ is g(x|λ) = 2φ(x) (λx), where φ(x) and (x) are N(0, 1) PDF and cumulative distribution function (CDF) respectively. Various extensions of SN(λ) have been proposed and studied (e.g., Arellano-Valle et al 2004;Arnold and Beaver 2002;Arnold et al 2007;Choudhury and Abdul 2011;Balakrishnan 2002;Gupta and Gupta 2004;Sharafi and Behboodian 2008;Yadegari et al 2008). For reviews on skewed normal and its generalization, one may refer to Kotz and Vicari (2005) and Lee et al (2013).…”

Section: Introductionmentioning

confidence: 99%

T-normal family of distributions: a new approach to generalize the normal distribution

Alzaatreh

Lee

Famoye

2014

J Stat Distrib Appl

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The idea of generating skewed distributions from normal has been of great interest among researchers for decades. This paper proposes four families of generalized normal distributions using the T-X framework. These four families of distributions are named as T-normal families arising from the quantile functions of (i) standard exponential, (ii) standard log-logistic, (iii) standard logistic and (iv) standard extreme value distributions. Some general properties including moments, mean deviations and Shannon entropy of the T-normal family are studied. Four new generalized normal distributions are developed using the T-normal method. Some properties of these four generalized normal distributions are studied in detail. The shapes of the proposed T-normal distributions can be symmetric, skewed to the right, skewed to the left, or bimodal. Two data sets, one skewed unimodal and the other bimodal, are fitted by using the generalized T-normal distributions.

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“…This approach underlies the general classes of skewed distributions generated, for example, by hidden truncation models (see e.g. Azzalini, 1985 andBeaver, 2002), inverse scale factors in the positive and negative orthants (Fernández and Steel, 1998) and order statistics (Jones, 2004).…”

Section: Introductionmentioning

confidence: 99%

A Constructive Representation of Univariate Skewed Distributions

Ferreira

Steel

2006

Journal of the American Statistical Association

118

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We introduce a general perspective on the introduction of skewness into symmetric distributions. Through inverse probability integral transformations we provide a constructive representation of skewed distributions, where the skewing mechanism and the original symmetric distributions are specified separately. We study the effects of the skewing mechanism on e.g. modality, tail behaviour and the amount of skewness generated. The representation is used to introduce novel classes of skewed distributions, where we induce certain prespecified characteristics through particular choices of the skewing mechanism. Finally, we use a Bayesian linear regression framework to compare the new classes with some existing distributions in the context of two empirical examples.

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Skewed multivariate models related to hidden truncation and/or selective reporting

Abstract: Skew-normal, skew-Cauchy, skew-Laplace, skew multivariate distributions, 62F03, 62A05,

Cited by 221 publications

References 16 publications

On describing multivariate skewed distributions: A directional approach

On describing multivariate skewed distributions: A directional approach

T-normal family of distributions: a new approach to generalize the normal distribution

A Constructive Representation of Univariate Skewed Distributions

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