Journal of Statistical Theory and Practice

, Volume 7, Issue 3, pp 480–495

# On Some Properties of the Unified Skew Normal Distribution

• Arjun K. Gupta
• Wei Ning
Article

## Abstract

Like some other multivariate skew normal distributions, the unified skew normal (SUN) distribution preserves important properties of the normal distribution. In this article, we show that for a random vector with a unified multivariate skew normal distribution all column (row) full rank linear transformations are in the same family of distributions. Using this property, we provide a characterization for the SUN distribution. In addition, we show that the joint distribution of the independent SUN random vectors is again a SUN distributed random vector. With this property and closure under the linear transformation, we finally show the closure of sums of independent SUN random vectors, and as expected it belongs to the same family.

## Keywords

Unified skew normal distribution Linear transformation Characterization Singular distribution Distribution of the sum

## AMS 2000 Subject Classification

Primary 62H10 Secondary 62P20

## References

1. Arellano-Valle, R. B., and A. Azzalini, 2006. On the unification of families of skew-normal distributions. Scand. J. Stat., 33, 561–574.
2. Arellano-Valle, R. B., and M. G. Genton. 2005. On fundamental skew distributions. J. Multivariate Anal., 96, 93–116.
3. Azzalini, A. 1985. A class of distributions which includes the normal ones. Scand. J. Stat., 12, 171–178.
4. Azzalini, A., and A. Capitanio. 1999. Statistical applications of the multivariate skew normal distributions. J. R. Stat. Soc. Ser. B, 61, 579–602.
5. Azzalini, A., and A. Dalla Valle. 1996. The multivariate skew-normal distribution. Biometrika, 83, 715–726.
6. Chen, J. T., A. K. Gupta, and C. G. Troskie. 2003. The distribution of stock returns when the market is up. Commun. Stat. Theory Methods, 32, 1541–1558.
7. Chen, J. T., A. K. Gupta, and T. T. Nguyen. 2004. The density of the skew normal sample mean and its applications. J. Stat. Comput. Simulation, 74, 487–494.
8. Dominguez-Monilla, J. A., G. Gonzalez-Farias, and A. K. Gupta. 2001. General multivariate skew normal distribution. Department of Mathematics and Statistics, Bowling Green State University. Technical Report no. 01-09.Google Scholar
9. Genton, M. G., ed. 2004. Skew-elliptical distributions and their applications: A journey beyond normality. Boca Raton, FL: Chapman and Hall/CRC.
10. Gonzalez-Farias, G., J. A. Dominguez-Molina, and A. K. Gupta. 2004. The closed skew-normal distribution. In Skew-elliptical distributions and their applications: A journey beyond normality, ed. M. G. Genton, 25–42. Boca Raton, FL: Chapman and Hall/CRC.Google Scholar
11. Gupta, A. K., and J. T. Chen. 2004. A class of multivariate skew-normal models. Ann. Inst. Stat. Math., 56(2), 305–315.
12. Gupta, A. K., G. Gonzalez-Farias, and J. A. Dominguez-Molina. 2004. A multivariate skew normal distribution. J. Multivariate Anal., 89, 181–190.
13. Gupta, A. K., and M. A. Aziz. 2011. Robust comonotonic lower convex order bound approximation for the distribution of terminal wealth in finance and actuarial science. Department of Mathematics and Statistics, Bowling Green State University, Technical Report, No. 11-13.Google Scholar
14. Gupta, A. K., and D. K. Nagar. 2000. Matrix variate distributions. Boca Raton, FL: Chapman and Hall/CRC.
15. Gupta, A. K., and T. Chen. 2001. Goodness-of-fit tests for the skew-normal distribution. Commun. Stat. Simulation Comput., 30, 907–930.
16. Gupta, A. K., and T. Kollo. 2003. Density expansions based on the multivariate skew normal distribution. Sankhya, 65, 821–835.
17. Gupta, A. K., and S. Nadarajah. 2007. Moments and cumulants of the skew normal distribution. Kobe J. Math., 24, 107–124.
18. Gupta, A. K., T. T. Nguyen, and J. A. T. Sanqui. 2004. Characterization of the skewnormal distribution. Ann. Inst. Statist. Math., 351–360.
19. Horn, R. A., and C. R. Johnson. 1991. Topics in matrix analysis. Cambridge, UK: Cambridge University Press.
20. Liseo, B., and N. Loperfido. 2003. A Bayesian interpretation of the multivariate skew-normal distribution. Stat. Probab. Lett., 61, 395–401.
21. Roberts, C. 1966. A correlation model useful in the study of twins. J. Am. Stat. Assoc., 61, 1184–1190.
22. Sahu, K., D. K. Dey, and M. D. Branco. 2003. A new class of multivariate skew distributions with applications to Bayesian regression models. Can. J. Stat., 31, 129–150.