Abstract
In this article, we consider a general form of univariate skewed distributions. We denote this form by GUS(λ; h(x)) or GUS with density s(x|λ, h(x)) = 2f(x)G(λ h(x)), where f is a symmetric density, G is a symmetric differentiable distribution, and h(x) is an odd function. A special case of this general form, normal case, is derived and denoted by GUSN(λ; h(x)). Some representations and some main properties of GUS(λ; h(x)) are studied. The moments of GUSN(λ; h(x)) and SN(λ), the known skew normal distribution of Azzalini (1985), are compared and the relationship between them is given. As an application, we use it to construct a new form for skew t-distribution and skew Cauchy distribution. In addition, we extend Stein’s lemma and study infinite divisibility of GUSN(λ; h(x)).
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References
Adcock CJ (2007) Extensions of Stein’s lemma for the skew-normal distribution. Commun Stat-Theory Methods 36: 1661–1671
Arellano-Valle RB, Genton MG (2005) On fundemental skew distributions. J Multivar Anal 96: 63–116
Arellano-Valle RB, Gomez HW, Quintana FA (2004) A new class of skew-normal distributions. Commun Stat-Theory Methods 33(7): 1465–1480
Arnold BC, Beaver RJ (2002) Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion). Test 11: 7–54
Azzalini A (1985) A class of distributions with includes the normal ones. Scand J Stat 12: 171–178
Azzalini A (2005) The skew-normal distribution and related multivariate families. Scand J Stat 32(2): 159–188
Azzalini A, Dalla-Valle A (1996) The multivariate skew-normal distribution. Biometrika 83(4): 715–726
Branco MD, Dey DK (2001) A general class of multivariate skew-elliptical distributions. J Multivar Anal 79: 99–113
Dominguez-Molina JA, Rocha-Artega A (2007) On the infinite divisibility of some skewed symmetric distributions. Stat Probab Lett 77: 644–648
Ferreira JTAS, Steel MFJ (2006) A constructive representation of univariate skewed distributions. J Am Stat Assoc 101: 823–829
Hamedani GG, Volkmer H (2006) An inverse problem for a class of continuous distributions with skewness. Stat Probab Lett 79(9): 945–951
Henze N (1986) A probabilistic representation of the skew-normal distribution. Scand J Stat 13: 271–275
Kozubowski TJ, Nolan JP (2007) Infinite divisibility of skew Gaussian and Laplace laws. Stat Probab Lett 78(6): 654–660
Steutel FW, Van Harn K (2003) Infinite divisibility of probability distributions on the real line. Marcel-Dekker, New York
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Abtahi, A., Behboodian, J. & Sharafi, M. A general class of univariate skew distributions considering Stein’s lemma and infinite divisibility. Metrika 75, 193–206 (2012). https://doi.org/10.1007/s00184-010-0322-8
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DOI: https://doi.org/10.1007/s00184-010-0322-8