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

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.

Stein's Lemma for the Reparameterization Trick with Exponential Family Mixtures

“…However, Price (1958); Opper & Archambeau (2009) use the characteristic function of Gaussian to prove Price's theorem, which is not easy to extend to the Gaussian mixture case. Hudson et al (1978); Brown (1986); Arnold et al (2001); Landsman (2006); Landsman & Nešlehová (2008); Kattumannil (2009); Kattumannil & Dewan (2016); Adcock (2007); Adcock & Shutes (2012) further extend Stein's lemma to exponential family and beyond. Unfortunately, these works neither show the connection between the gradient identity and the implicit reparameterization trick nor give any second-order identity.…”

Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

“…The normal gamma distribution fulfills the assumptions of Lemma 4, when W is chosen to be a d-dimensional normal vector with mean μ and variance , while U is a gamma random variable with scale parameter 1 and shape parameter υ; see Adcock and Shutes (2012). The normal gamma distribution fulfills the assumptions of Lemma 4, when W is chosen to be a d-dimensional normal vector with mean μ and variance , while U is a gamma random variable with scale parameter 1 and shape parameter υ; see Adcock and Shutes (2012).…”

Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

“…The normal gamma distribution fulfills the assumptions of Lemma 4, when W is chosen to be a d-dimensional normal vector with mean μ and variance , while U is a gamma random variable with scale parameter 1 and shape parameter υ; see Adcock and Shutes (2012).…”