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

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

\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),

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

Properties

Joint moment generating function

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

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

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

{\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 $Y_{i}$ is as the following:

\operatorname{E}(Y_{i})=\frac{1}{\mu_i}\big[\ln(\lambda_i/\delta)+\digamma(\nu)\big],

\operatorname{var}(Z_i)=\digamma^{[1]}(\nu)/(\mu_i)^2

where $\digamma(\nu)$ and $\digamma^{[1]}(\nu)$ are values of digamma and trigamma functions at $\nu$ , 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 $\boldsymbol{T}\sim \mathrm{G}\text{-}\mathrm{MVGB}(\delta,\nu,\boldsymbol{\lambda},\boldsymbol{\mu})$ is the following:

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

↑ 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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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