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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. Adcock
  • K. Shutes
Article

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.

Keywords

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

AMS Subject Classification

62E15 62H05 62P05 

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Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

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

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