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

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Abstract

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.

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References

  • Abramowitz, M., and I. Stegun. 1965. Handbook of mathematical functions. New York, Dover.

    MATH  Google Scholar 

  • 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.

    MATH  Google Scholar 

  • 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 

  • Adcock, C. J. 2005. Exploiting skewness to build an optimal hedge fund with a currency overlay. Eur. J. Finance, 11, 445–462.

    Article  Google Scholar 

  • Adcock, C. J. 2007. Extensions of Stein’s lemma for the skew-normal distribution. Commun. Stat. Theory Methods, 36, 1661–1672.

    Article  MathSciNet  Google Scholar 

  • Adcock, C. J., 2010. Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution. Ann. Operations Res., 176, 221–234.

    Article  MathSciNet  Google Scholar 

  • 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.

  • 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 

  • Adcock, C. J., and K. Shutes. 2005. An analysis of skewness and skewness persistence in three emerging markets, Emerging Markets Rev., 6, 396–418.

    Article  Google Scholar 

  • Aigner, D. J., C. K. Lovell, and P. Schmidt. 1977. Formulation and estimation of stochastic production function model. J. of Econometrics, 12, 21–37.

    Article  MathSciNet  Google Scholar 

  • Arnold, B. C., and R. J. Beaver. 2000. Hidden truncation models. Sankhya Ser. A, 62, 22–35.

    MathSciNet  MATH  Google Scholar 

  • Azzalini, A. 1985. A class of distributions which includes the normal ones. Scan. J. Stat., 12, 171–178.

    MathSciNet  MATH  Google Scholar 

  • Azzalini, A. 1986. Further results on a class of distributions which includes the normal ones. Statistica, 46, 199–208.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Azzalini, A., and A. Dalla Valle. 1996. The multivariate skew-normal distribution. Biometrika, 83, 715–726.

    Article  MathSciNet  Google Scholar 

  • Badrinath, S. G., and S. Chatterjee. 1988. On measuring skewness and elongation in common stock return distributions. J. Business, 61, 451–472.

    Article  Google Scholar 

  • Beedles, W. L., 1979. On the asymmetry of market returns, J. Financial Quan. Anal., 14, 653–660.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Branco, M. D., and D. K. Dey. 2001. A general class of multivariate skew-elliptical distributions. J. Multivariate Anal., 79, 99–113.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Corns, T. R. A. and S. E. Satchell. 2007. Skew Brownian motion and pricing European options. Eur. J. Finance, 13, 523–544.

    Article  Google Scholar 

  • Dey, D. K., and J. Liu. 2004. Prior elicitation from expert opinion: An interactive approach. University of Connecticut Division of Biostatistics, Working Paper.

  • 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.

    Article  MathSciNet  Google Scholar 

  • Fama, E. 1970. Efficient capital markets: A review of theory and empirical work. J. Finance, 25, 383–417.

    Article  Google Scholar 

  • Genton, M. 2004. Skew elliptical distributions and their applications: A journey beyond normality. Boca Raton, FL: Chapman and Hall.

    Book  Google Scholar 

  • Greene, W. H. 1990. A gamma-distributed stochastic frontier model. J. Econometrics, 46, 141–163.

    Article  MathSciNet  Google Scholar 

  • Harvey, C. R., and A. Siddique. 1997. Conditional skewness in asset pricing tests. J. Finance, 55, 1263–1295.

    Article  Google Scholar 

  • Harvey, C. R., J. C. Leichty, M. W. Leichty, and P. Muller. 2010. Portfolio selection with higher moments. Quant. Finance, 10, 469–485.

    Article  MathSciNet  Google Scholar 

  • Johnson, N., and S. Kotz. 1970. Continuous univariate distributions 1. Boston, Wiley.

    MATH  Google Scholar 

  • Kattumannil, S. K., 2009. On Stein’s identity and its applications. Stat. Probability Lett. 79, 1444–1449.

    Article  MathSciNet  Google Scholar 

  • Kraus, A., and R. H. Litzenberger. 1976. Skewness preference and the valuation of risk assets. J. Finance, 31, 1085–1100.

    Google Scholar 

  • Landsman, Z., and J. Nešlehová. 2008. Stein’s Lemma for elliptical random vectors. J. Multivariate Anal., 99, 912–927

    Article  MathSciNet  Google Scholar 

  • Liseo, B., and N. Loperfido. 2003. A Bayesian interpretation of the multivariate skew-normal distribution. Stat. Probability Lett., 61(4), 395–401.

    Article  MathSciNet  Google Scholar 

  • Liu, J. S. 1994. Siegel’s formula via Stein’s identities. Stat. and Probability Lett., 21, 247–251.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Shutes, K. 2005. Non-normality in asset pricing—Extensions and applications of the skew-normal distribution. PhD Thesis. University of Sheffield.

  • Siegel, A. F. 1993. A surprising covariance involving the minimum of multivariate normal variables. J. Am. Stat. Assoc., 88, 77–80.

    MathSciNet  MATH  Google Scholar 

  • Simaan, Y. 1993. Portfolio selection and asset pricing—Three parameter framework. Manage. Sci., 39, 568–587.

    Article  Google Scholar 

  • Stein, C. 1981. Estimation of the mean of a multivariate normal distribution. Ann. Stat., 9, 1135–1151.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Management School, University of Sheffield, 9, Mappin Street, Sheffield, S1 4DT, UK

    C. J. Adcock

  2. LEI, Wageningen University, The Hague, The Netherlands

    K. Shutes

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  1. C. J. Adcock
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  2. K. Shutes
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Correspondence to C. J. Adcock.

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Adcock, C.J., Shutes, K. On the Multivariate Extended Skew-Normal, Normal-Exponential, and Normal-Gamma Distributions. J Stat Theory Pract 6, 636–664 (2012). https://doi.org/10.1080/15598608.2012.719799

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  • DOI: https://doi.org/10.1080/15598608.2012.719799

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