Abstract
The skew normal distribution of Azzalini (Scand J Stat 12:171–178, 1985) has been found suitable for unimodal density but with some skewness present. Through this article, we introduce a flexible extension of the Azzalini (Scand J Stat 12:171–178, 1985) skew normal distribution based on a symmetric component normal distribution (Gui et al. in J Stat Theory Appl 12(1):55–66, 2013). The proposed model can efficiently capture the bimodality, skewness and kurtosis criteria and heavy-tail property. The paper presents various basic properties of this family of distributions and provides two stochastic representations which are useful for obtaining theoretical properties and to simulate from the distribution. Further, maximum likelihood estimation of the parameters is studied numerically by simulation and the distribution is investigated by carrying out comparative fitting of three real datasets.
Similar content being viewed by others
References
Arellano-Valle RB, Cortes MA, Gomez HW (2010) An extension of the epsilon-skew-normal distribution. Commun Stat Theory Methods 39(5):912–922
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, Gomez HW, Salinas HS (2014) A doubly skewed normal distribution. Stat J Theor App Stat 49(4):842–858
Azzalini A (1985) A class of distributions which include the normal. Scand J Stat 12:171–178
Azzalini A (1986) Further results on a class of distributions which includes the normal ones. Statistica 46:199–208
Azzalini A, Bowman AW (1990) A look at some data on the Old Faithful geyser. J R Stat Soc Ser C 39(3):357–365
Azzalini A, Regoli G (2012) Some properties of skew-symmetric distributions. Ann Inst Stat Math 64(4):857–879
Burnham KP, Anderson DR (2002) Model selection and multimodel inference: a practical information-theoretic approach, 2nd edn. Springer, New York
Chiogna M (2005) A note on the asymptotic distribution of the maximum likelihood estimator for the scalar skew-normal distribution. Stat Methods Appl 14(3):331–341
Evans DL, Drew JH, Leemis LM (2008) The distribution of the Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling test statistics for exponential populations with estimated parameters. Commun Stat Simul Comput 37(7):1396–1421
Elal-Olivero D (2010) Alpha-skew-normal distribution. Proyecc J Math 29(3):224–240
Ferreira JTAS, Steel MFJ (2006) A constructive representation of univariate skewed distributions. J Am Stat Assoc 101(474):823–829
Fernandez C, Steel MFJ (1998) On Bayesian modeling of fat tails and skewness. J Am Stat Assoc 93(441):359–371
Gomez HW, Elal-Olivero D, Salinas HS, Bolfarine H (2011) Bimodal extension based on the skew-normal distribution with application to pollen data. Environmetrics 22(1):50–62
Gupta AK, Chen T (2001) Goodness of fit tests for the skew-normal distribution. Commun Stat Simul Comput 30(4):907–930
Gui W, Chen PH, Wu H (2013) A Symmetric Component Alpha Normal Slash Distribution: Properties and Inferences. J Stat Theory Appl 12(1):55–66
Henze N (1986) A probabilistic representation of the skew-normal distribution. Scand J Stat 13(4):271–275
Kim HJ (2005) On a class of two-piece skew-normal distributions. Stat J Theor Appl Stat 39(6):537–553
Liseo B (1990) La classe delle densita normali sghembe: aspetti inferenziali da un punto di vista bayesiano. Statistica 50(1):71–82
Ma Y, Genton MG (2004) Flexible class of skew-symmetric distributions. Scand J Stat 31(3):459–468
Mameli V, Musio M (2013) A generalization of the skew-normal distribution: the beta skew-normal. Commun Stat Theory Methods 42(12):2229–2244
Martinez EH, Varela H, Gomez HW, Bolfarine H (2008) A note on the likelihood and moments of the skew-normal distribution. Stat Oper Res Trans 32(1):57–66
Mudholkar GS, Hutson AD (2000) The epsilon-skew-normal distribution for analyzing near normal data. J Stat Plan Inference 83(2):291–309
Nekoukhou V, Alamatsaz MH, Aghajani AH (2013) A flexible skew-generalized normal distribution. Commun Stat Theory Methods 42(13):2324–2334
Owen DB (1956) Tables for computing bivariate normal probabilities. Ann Math Stat 27(4):1075–1090
Pewsey A (2000) Problems of inference for Azzalini’s skew-normal distribution. J Appl Stat 27(7):859–870
Roberts HV (1988) Data analysis for managers with Minitab. Scientific Press, Redwood City
Acknowledgments
The authors acknowledge helpful comments and suggestions from three referees that substantially improved the presentation.
Author information
Authors and Affiliations
Corresponding author
Appendix 1
Appendix 1
1.1 Appendix A.1: score vector and Hessian matrix
Let \(x_1 , x_2 ,\ldots , x_n\) is a random sample drawn from the skew symmetric component normal distribution \(SSCN(\mu ,\sigma ,\lambda ,\alpha )\), then the log-likelihood function is given by (13). The elements of the score vector are obtained by differentiation
where \(z_i = \frac{{x_i - \mu }}{\sigma }\) and \(w_i (a) = \sum \limits _{i = 1}^n {z_i^a \frac{{\phi (\lambda z_i )}}{{\varPhi (\lambda z_i )}}}\).
The Hessian matrix, second partial derivatives of the log-likelihood, is given by
where
Rights and permissions
About this article
Cite this article
Rasekhi, M., Chinipardaz, R. & Alavi, S.M.R. A flexible generalization of the skew normal distribution based on a weighted normal distribution. Stat Methods Appl 25, 375–394 (2016). https://doi.org/10.1007/s10260-015-0337-4
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10260-015-0337-4