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 ^[1] 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 ${\displaystyle {\boldsymbol {Y}}\sim \mathrm {G} {\text{-}}\mathrm {MVLG} (\delta ,\nu ,{\boldsymbol {\lambda }},{\boldsymbol {\mu }})}$, the joint probability density function (pdf) of ${\displaystyle {\boldsymbol {Y}}=(Y_{1},\dots ,Y_{k})}$ is given as the following:

${\displaystyle f(y_{1},\dots ,y_{k})=\delta ^{\nu }\sum _{n=0}^{\infty }{\frac {(1-\delta )^{n}\prod _{i=1}^{k}\mu _{i}\lambda _{i}^{-\nu -n}}{[\Gamma (\nu +n)]^{k-1}\Gamma (\nu )n!}}\exp {\bigg \{}(\nu +n)\sum _{i=1}^{k}\mu _{i}y_{i}-\sum _{i=1}^{k}{\frac {1}{\lambda _{i}}}\exp\{\mu _{i}y_{i}\}{\bigg \}},}$

where ${\displaystyle {\boldsymbol {y}}\in \mathbb {R} ^{k},\nu >0,\lambda _{j}>0,\mu _{j}>0}$ for ${\displaystyle j=1,\dots ,k,\delta =\det({\boldsymbol {\Omega }})^{\frac {1}{k-1}},}$ and

${\displaystyle {\boldsymbol {\Omega }}=\left({\begin{array}{cccc}1&{\sqrt {\mathrm {abs} (\rho _{12})}}&\cdots &{\sqrt {\mathrm {abs} (\rho _{1k})}}\\{\sqrt {\mathrm {abs} (\rho _{12})}}&1&\cdots &{\sqrt {\mathrm {abs} (\rho _{2k})}}\\\vdots &\vdots &\ddots &\vdots \\{\sqrt {\mathrm {abs} (\rho _{1k})}}&{\sqrt {\mathrm {abs} (\rho _{2k})}}&\cdots &1\end{array}}\right),}$

${\displaystyle \rho _{ij}}$ is the correlation between ${\displaystyle Y_{i}}$ and ${\displaystyle Y_{j}}$, ${\displaystyle \det(\cdot )}$ and ${\displaystyle \mathrm {abs} (\cdot )}$ denote determinant and absolute value of inner expression, respectively, and ${\displaystyle {\boldsymbol {g}}=(\delta ,\nu ,{\boldsymbol {\lambda }}^{T},{\boldsymbol {\mu }}^{T})}$ includes parameters of the distribution.

Properties

Joint moment generating function

The joint moment generating function of G-MVLG distribution is as the following:

${\displaystyle M_{\boldsymbol {Y}}({\boldsymbol {t}})=\delta ^{\nu }{\bigg (}\prod _{i=1}^{k}\lambda _{i}^{t_{i}/\mu _{i}}{\bigg )}\sum _{n=0}^{\infty }{\frac {\Gamma (\nu +n)}{\Gamma (\nu )n!}}(1-\delta )^{n}\prod _{i=1}^{k}{\frac {\Gamma (\nu +n+t_{i}/\mu _{i})}{\Gamma (\nu +n)}}.}$

Marginal central moments

${\displaystyle r^{\text{th}}}$ marginal central moment of ${\displaystyle Y_{i}}$ is as the following:

${\displaystyle {\mu _{i}}'_{r}=\left[{\frac {(\lambda _{i}/\delta )^{t_{i}/\mu _{i}}}{\Gamma (\nu )}}\sum _{k=0}^{r}{\binom {r}{k}}\left[{\frac {\ln(\lambda _{i}/\delta )}{\mu _{i}}}\right]^{r-k}{\frac {\partial ^{k}\Gamma (\nu +t_{i}/\mu _{i})}{\partial t_{i}^{k}}}\right]_{t_{i}=0}.}$

Marginal expected value and variance

Marginal expected value ${\displaystyle Y_{i}}$ is as the following:

${\displaystyle \operatorname {E} (Y_{i})={\frac {1}{\mu _{i}}}{\big [}\ln(\lambda _{i}/\delta )+\digamma (\nu ){\big ]},}$

${\displaystyle \operatorname {var} (Z_{i})=\digamma ^{[1]}(\nu )/(\mu _{i})^{2}}$

where ${\displaystyle \digamma (\nu )}$ and ${\displaystyle \digamma ^{[1]}(\nu )}$ are values of digamma and trigamma functions at ${\displaystyle \nu }$, respectively.

Related distributions

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 ${\displaystyle {\boldsymbol {T}}\sim \mathrm {G} {\text{-}}\mathrm {MVGB} (\delta ,\nu ,{\boldsymbol {\lambda }},{\boldsymbol {\mu }})}$ is the following:

${\displaystyle f(t_{1},\dots ,t_{k};\delta ,\nu ,{\boldsymbol {\lambda }},{\boldsymbol {\mu }}))=\delta ^{\nu }\sum _{n=0}^{\infty }{\frac {(1-\delta )^{n}\prod _{i=1}^{k}\mu _{i}\lambda _{i}^{-\nu -n}}{[\Gamma (\nu +n)]^{k-1}\Gamma (\nu )n!}}\exp {\bigg \{}-(\nu +n)\sum _{i=1}^{k}\mu _{i}t_{i}-\sum _{i=1}^{k}{\frac {1}{\lambda _{i}}}\exp\{-\mu _{i}t_{i}\}{\bigg \}},\quad t_{i}\in \mathbb {R} .}$

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..

References

1. Demirhan, Haydar; Hamurkaroglu, Canan (2011). "On a multivariate log-gamma distribution and the use of the distribution in the Bayesian analysis". Journal of Statistical Planning and Inference. 141 (3): 1141–1152. doi:10.1016/j.jspi.2010.09.015.