Journal of Statistical Theory and Practice

, Volume 6, Issue 4, pp 636–664 | Cite as

On the Multivariate Extended Skew-Normal, Normal-Exponential, and Normal-Gamma Distributions

  • C. J. AdcockEmail author
  • K. Shutes


This paper presents expressions for the multivariate normal-exponential and normal-gamma distributions. It then presents properties of these distributions. These include conditional distributions and a new extension to Stein’s lemma. It is also shown that the multivariate normal-gamma and normal-exponential distribution are not in general closed under conditioning, although they are closed under linear transformations. The paper also demonstrates that there are relationships between the extended skew-normal distribution and the normal-gamma and normal-exponential distributions. Specifically, it is shown that certain limiting cases of the extended skew-normal distribution are normal-gamma and normal-exponential. One interpretation of these results is that the normal-exponential distribution may be considered to be an alternative model to the extended skew-normal in some situations. An alternative point of view, however, is that the normal-exponential distribution is redundant since it may be replicated by a suitable extended skew-normal distribution. The theoretical results are supported by an empirical study of stock returns, which includes use of the multivariate distributions for portfolio selection.


Exponential distribution Gamma distribution Moment-generating function Multivariate extended skew-normal distribution Portfolio selection Stein’s lemma 

AMS Subject Classification

62E15 62H05 62P05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramowitz, M., and I. Stegun. 1965. Handbook of mathematical functions. New York, Dover.zbMATHGoogle Scholar
  2. Adcock, C. J. 2002. Asset pricing and portfolio selection based on the multivariate skew-student distribution. Multi-Moment Capital Asset Pricing Models and Related Topics Workshop, Paris.zbMATHGoogle Scholar
  3. Adcock, C. J. 2004. Capital asset pricing for UK stocks under the multivariate skew-normal distribution. In Skew Elliptical distributions and their applications: A journey beyond normality, ed. M. Genton, 191–204. Boca Raton, Chapman and Hall.Google Scholar
  4. Adcock, C. J. 2005. Exploiting skewness to build an optimal hedge fund with a currency overlay. Eur. J. Finance, 11, 445–462.CrossRefGoogle Scholar
  5. Adcock, C. J. 2007. Extensions of Stein’s lemma for the skew-normal distribution. Commun. Stat. Theory Methods, 36, 1661–1672.MathSciNetCrossRefGoogle Scholar
  6. Adcock, C. J., 2010. Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution. Ann. Operations Res., 176, 221–234.MathSciNetCrossRefGoogle Scholar
  7. Adcock, C. J., N. Areal, M. J. R. Armada, M. C. Cortez, B. Oliveira, and F. Silva. 2012. Conditions under which portfolio performance measures are monotonic functions of the Sharpe ratio. University of Miaho, Working paper.Google Scholar
  8. Adcock, C. J., and K. Shutes. 2001. Portfolio selection based on the multivariate-skew normal distribution, In Financial modelling, ed. A. Skulimowski, 167–177. Krakow, Progress & Business Publishers.Google Scholar
  9. Adcock, C. J., and K. Shutes. 2005. An analysis of skewness and skewness persistence in three emerging markets, Emerging Markets Rev., 6, 396–418.CrossRefGoogle Scholar
  10. Aigner, D. J., C. K. Lovell, and P. Schmidt. 1977. Formulation and estimation of stochastic production function model. J. of Econometrics, 12, 21–37.MathSciNetCrossRefGoogle Scholar
  11. Arnold, B. C., and R. J. Beaver. 2000. Hidden truncation models. Sankhya Ser. A, 62, 22–35.MathSciNetzbMATHGoogle Scholar
  12. Azzalini, A. 1985. A class of distributions which includes the normal ones. Scan. J. Stat., 12, 171–178.MathSciNetzbMATHGoogle Scholar
  13. Azzalini, A. 1986. Further results on a class of distributions which includes the normal ones. Statistica, 46, 199–208.MathSciNetzbMATHGoogle Scholar
  14. Azzalini, A. 2005. The skew-normal distribution and related multivariate families (with discussion by Marc G. Genton and a rejoinder by the author). Scand. J. Stat., 32, 159–200.CrossRefGoogle Scholar
  15. Azzalini, A., and A. Capitanio. 2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. R. Stat. Soc. Ser. B, 65, 367–389.MathSciNetCrossRefGoogle Scholar
  16. Azzalini, A., and A. Dalla Valle. 1996. The multivariate skew-normal distribution. Biometrika, 83, 715–726.MathSciNetCrossRefGoogle Scholar
  17. Badrinath, S. G., and S. Chatterjee. 1988. On measuring skewness and elongation in common stock return distributions. J. Business, 61, 451–472.CrossRefGoogle Scholar
  18. Beedles, W. L., 1979. On the asymmetry of market returns, J. Financial Quan. Anal., 14, 653–660.CrossRefGoogle Scholar
  19. Bekaert G., C. R. Harvey, C. B. Erb, and T. E. Viskantam. 1998. Distributional characteristics of emerging market returns & asset allocation. J. Portfolio Manage., 24, 102–116.CrossRefGoogle Scholar
  20. Branco, M. D., and D. K. Dey. 2001. A general class of multivariate skew-elliptical distributions. J. Multivariate Anal., 79, 99–113.MathSciNetCrossRefGoogle Scholar
  21. Chunhachinda, P., K. Dandapani, S. Hamid, and A. J. Prakash. 1997. Portfolio selection and skewness: Evidence from international stock markets. J. Banking Finance, 21, 143–167.CrossRefGoogle Scholar
  22. Corns, T. R. A. and S. E. Satchell. 2007. Skew Brownian motion and pricing European options. Eur. J. Finance, 13, 523–544.CrossRefGoogle Scholar
  23. Dey, D. K., and J. Liu. 2004. Prior elicitation from expert opinion: An interactive approach. University of Connecticut Division of Biostatistics, Working Paper.Google Scholar
  24. Elal-Oliveroa, D., H. W. Gómez, and F. A. Quintanac. 2009. Bayesian modeling using a class of bimodal skew-elliptical distributions. J. Stat. Plan. Inference, 139, 1484–1492.MathSciNetCrossRefGoogle Scholar
  25. Fama, E. 1970. Efficient capital markets: A review of theory and empirical work. J. Finance, 25, 383–417.CrossRefGoogle Scholar
  26. Genton, M. 2004. Skew elliptical distributions and their applications: A journey beyond normality. Boca Raton, FL: Chapman and Hall.CrossRefGoogle Scholar
  27. Greene, W. H. 1990. A gamma-distributed stochastic frontier model. J. Econometrics, 46, 141–163.MathSciNetCrossRefGoogle Scholar
  28. Harvey, C. R., and A. Siddique. 1997. Conditional skewness in asset pricing tests. J. Finance, 55, 1263–1295.CrossRefGoogle Scholar
  29. Harvey, C. R., J. C. Leichty, M. W. Leichty, and P. Muller. 2010. Portfolio selection with higher moments. Quant. Finance, 10, 469–485.MathSciNetCrossRefGoogle Scholar
  30. Johnson, N., and S. Kotz. 1970. Continuous univariate distributions 1. Boston, Wiley.zbMATHGoogle Scholar
  31. Kattumannil, S. K., 2009. On Stein’s identity and its applications. Stat. Probability Lett. 79, 1444–1449.MathSciNetCrossRefGoogle Scholar
  32. Kraus, A., and R. H. Litzenberger. 1976. Skewness preference and the valuation of risk assets. J. Finance, 31, 1085–1100.Google Scholar
  33. Landsman, Z., and J. Nešlehová. 2008. Stein’s Lemma for elliptical random vectors. J. Multivariate Anal., 99, 912–927MathSciNetCrossRefGoogle Scholar
  34. Liseo, B., and N. Loperfido. 2003. A Bayesian interpretation of the multivariate skew-normal distribution. Stat. Probability Lett., 61(4), 395–401.MathSciNetCrossRefGoogle Scholar
  35. Liu, J. S. 1994. Siegel’s formula via Stein’s identities. Stat. and Probability Lett., 21, 247–251.MathSciNetCrossRefGoogle Scholar
  36. Samuelson, P. A. 1970. The fundamental application theorem of portfolio analysis in terms of means, variances and higher moments. Rev. Econ. Stud., 37, 537–542.CrossRefGoogle Scholar
  37. Shutes, K. 2005. Non-normality in asset pricing—Extensions and applications of the skew-normal distribution. PhD Thesis. University of Sheffield.Google Scholar
  38. Siegel, A. F. 1993. A surprising covariance involving the minimum of multivariate normal variables. J. Am. Stat. Assoc., 88, 77–80.MathSciNetzbMATHGoogle Scholar
  39. Simaan, Y. 1993. Portfolio selection and asset pricing—Three parameter framework. Manage. Sci., 39, 568–587.CrossRefGoogle Scholar
  40. Stein, C. 1981. Estimation of the mean of a multivariate normal distribution. Ann. Stat., 9, 1135–1151.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.The Management SchoolUniversity of SheffieldSheffieldUK
  2. 2.LEIWageningen UniversityThe HagueThe Netherlands

Personalised recommendations