The Multivariate Alpha Skew Gaussian Distribution


In this paper we propose a new class of probability distributions, so called multivariate alpha skew normal distribution. It can accommodate up to two modes and generalizes the distribution proposed by Elal-Olivero [Proyecciones (Antofagasta) 29(3):224–240, 2010] in its marginal components. Its properties are studied. In particular, we derive its standard and non-standard densities, moment generating functions, expectations, variance-covariance matrixes, marginal and conditional distributions. Estimation is based on maximum likelihood. The asymptotic properties of the inferential procedure are verified in the light of a simulation study. The usefulness of the new distribution is illustrated in a real benchmark data.

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  1. Arellano-Valle, R.B., Azzalini, A.: On the unification of families of skew-normal distributions. Scand. J. Stat. 33(3), 561–574 (2006)

    MathSciNet  Article  Google Scholar 

  2. Arellano-Valle, R.B., Cortes, M.A., Gomez, H.W.: An extension of the epsilon skew normal distribution. Commun. Stat. Theory Methods 39(5), 912–922 (2010)

    MathSciNet  Article  Google Scholar 

  3. Arellano-Valle, R.B., Genton, M.G., Loschi, R.H.: Shape mixtures of multivariate skew-normal distributions. J. Multivar. Anal. 100(1), 91–101 (2009)

    MathSciNet  Article  Google Scholar 

  4. Arellano-Valle, R.B., Gómez, H.W., Quintana, F.A.: A new class of skew normal distributions. Commun. Stat. Theory Methods 33(7), 1465–1480 (2004)

    MathSciNet  Article  Google Scholar 

  5. Azzalini, A.: A class of distributions which includes the normal ones. Scand. J. Stat. 12(2), 171–178 (1985)

    MathSciNet  MATH  Google Scholar 

  6. Azzalini, A., Capitanio, A.: Statistical applications of the multivariate skew normal distribution. J. R. Stat. Soc.: Ser. B (Stat. Methodol.) 61(3), 579–602 (1999)

    MathSciNet  Article  Google Scholar 

  7. Azzalini, A., Valle, A.D.: The multivariate skew-normal distribution. Biometrika 83(4), 715–726 (1996)

    MathSciNet  Article  Google Scholar 

  8. Bahrami, W., Agahi, H., Rangin, H.: A two-parameter balakrishnan skew-normal distribution. J. Stat. Res. Iran 6, 231–242 (2009)

    Google Scholar 

  9. Branco, M.D., Dey, D.K.: A general class of multivariate skew-elliptical distributions. J. Multivar. Anal. 79(1), 99–113 (2001)

    MathSciNet  Article  Google Scholar 

  10. da Silva Ferreira, C., Bolfarine, H., Lachos, V.H.: Skew scale mixtures of normal distributions: properties and estimation. Stat. Methodol. 8(2), 154–171 (2011)

    MathSciNet  Article  Google Scholar 

  11. Elal-Olivero, D.: Alpha-skew-normal distribution. Proyecciones (Antofagasta) 29(3), 224–240 (2010)

    MathSciNet  Article  Google Scholar 

  12. Everitt, B.S., Hothorn, T.: Maintainer Torsten Hothorn, and Chapman Everitt. Package HSAUR3 (2014)

  13. Ferreira, C.S., Lachos, V.H., Bolfarine, H.: Likelihood-based inference for multivariate skew scale mixtures of normal distributions. AStA Adv. Stat. Anal. 100(4), 421–441 (2016)

    MathSciNet  Article  Google Scholar 

  14. Gómez, H.W., Salinas, H.S., Bolfarine, H.: Generalized skew-normal models: properties and inference. Statistics 40(6), 495–505 (2006)

    MathSciNet  Article  Google Scholar 

  15. Gui, W., Chen, P.-H., Haiyan, W.: A symmetric component alpha normal slash distribution: properties and inferences. J. Stat. Theory Appl. 12(1), 55–66 (2012)

    MathSciNet  Google Scholar 

  16. Gupta, A.K., González-Farías, G., Domínguez-Molina, J.A.: A multivariate skew normal distribution. J. Multivar. Anal. 89(1), 181–190 (2004)

    MathSciNet  Article  Google Scholar 

  17. Handam, A.H.: A note on generalized alpha-skew-normal distribution. Int. J. Pure Appl. Math. 74(4), 491–496 (2012)

    MATH  Google Scholar 

  18. Harandi, S.S., Alamatsaz, M.H.: Alpha-skew-laplace distribution. Stat. Probab. Lett. 83, 774–782 (2013)

    MathSciNet  Article  Google Scholar 

  19. Hothorn, T., Everitt, B.S.: A Handbook of Statistical Analyses Using R. CRC Press, Boca Raton (2014)

    Google Scholar 

  20. Jamalizadeh, A., Behboodian, J., Balakrishnan, N.: A two-parameter generalized skew-normal distribution. Stat. Prob. Lett. 78(13), 1722–1726 (2008)

    MathSciNet  Article  Google Scholar 

  21. Kotecha, J.H., Djuric, P.M.: Gibbs sampling approach for generation of truncated multivariate gaussian random variables. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, 1999, Proceedings, vol. 3, pp. 1757–1760 (1999)

  22. Louzada, F., Ara, A., Fernandes, G.: The bivariate alpha-skew-normal distribution. Commun. Stat.-Theory Methods (2016) (just-accepted)

  23. Mahalanobis, P.C.: On the Generalized Distance in Statistics. In: Proceedings of National Institute of Sciences (India), vol. 2, pp. 49–55 (1936)

  24. Mardia, K.V.: Measures of multivariate skewness and kurtosis with applications. Biometrika 57(3), 519–530 (1970)

    MathSciNet  Article  Google Scholar 

  25. Mayor, M., Frei, P.-Y.: New worlds in the cosmos: the discovery of exoplanets. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  26. McAssey, M.P.: An empirical goodness-of-fit test for multivariate distributions. J. Appl. Stat. 40(5), 1120–1131 (2013)

    MathSciNet  Article  Google Scholar 

  27. R Core Team: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2013)

  28. Seber, G.A.F.: Linear Regression Analysis. Wiley, New York (1977)

  29. Stewart, G.W.: Matrix Algorithms: Basic Decompositions. SIAM (Society for industrial and applied mathematics), Philadelphia (1998)

    Google Scholar 

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The authors thank the reviewers for their comments and suggestions, which led to a substantial improvement of the manuscript. The research was partial sponsored by the Brazilian organizations CNPq ans FAPESP through their research grant programs.

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Ara, A., Louzada, F. The Multivariate Alpha Skew Gaussian Distribution. Bull Braz Math Soc, New Series 50, 823–843 (2019).

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  • Alpha skew Gaussian distribution
  • Asymmetry
  • Bimodality
  • Multivariate distribution