Generalized multivariate log-gamma distribution
In probability theory and statistics, the generalized multivariate log-gamma (G-MVLG) distribution is a multivariate distribution introduced by Demirhan and Hamurkaroglu in 2011. The G-MVLG is a flexible distribution. Skewness and kurtosis are well controlled by the parameters of the distribution. This enables one to control dispersion of the distribution. Because of this property, the distribution is effectively used as a joint prior distribution in Bayesian analysis, especially when the likelihood is not from the location-scale family of distributions such as normal distribution.
Joint probability density function
If , the joint probability density function (pdf) of is given as the following:
where for and
Joint moment generating function
The joint moment generating function of G-MVLG distribution is as the following:
Marginal central moments
marginal central moment of is as the following:
Marginal expected value and variance
Marginal expected value is as the following:
Demirhan and Hamurkaroglu establish a relation between the G-MVLG distribution and the Gumbel distribution (type I extreme value distribution) and gives a multivariate form of the Gumbel distribution, namely the generalized multivariate Gumbel (G-MVGB) distribution. The joint probability density function of is the following:
The Gumbel distribution has a broad range of applications in the field of risk analysis. Therefore, the G-MVGB distribution should be beneficial when it is applied to these types of problems..