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In [[probability theory]] and [[statistics]], the '''generalized multivariate log-gamma (G-MVLG) distribution''' is a [[multivariate distribution]] introduced by Demirhan and Hamurkaroglu<ref>{{cite journal|last=Demirhan|first=Haydar|coauthors=Hamurkaroglu, Canan|title=On a multivariate log-gamma distribution and the use of the distribution in the Bayesian analysis|journal=Journal of Statistical Planning and Inference|year=2011|volume=141|pages=1141–1152|doi=10.1016/j.jspi.2010.09.015|issue=3}}</ref> 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 [[statistical dispersion|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]].
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==Joint probability density function==
If <math>\boldsymbol{Y} \sim
\mathrm{G}\text{-}\mathrm{MVLG}(\delta,\nu,\boldsymbol{\lambda},\boldsymbol{\mu})</math>, the joint [[probability density function]] (pdf) of <math>\boldsymbol{Y}=(Y_{1},\dots,Y_{k})</math> is given as the following:
 
:<math>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\},</math>
 
where <math> \boldsymbol{y}\in \mathbb{R}^{k}, \nu>0, \lambda_{j}>0, \mu_{j}>0</math> for <math>j=1,\dots,k, \delta=\det(\boldsymbol{\Omega})^{\frac{1}{k-1}},</math> and
 
: <math>
 
\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),  
</math>
 
<math>\rho_{ij}</math> is the [[correlation]] between <math>Y_i</math> and <math>Y_j</math>, <math> \det(\cdot)</math> and <math>\mathrm{abs}(\cdot)</math> denote [[determinant]] and [[absolute value]] of inner expression, respectively, and <math>\boldsymbol{g}=(\delta,\nu,\boldsymbol{\lambda}^T,\boldsymbol{\mu}^T)</math> includes parameters of the distribution.
 
==Properties==
===Joint moment generating function===
 
The joint [[moment generating function]] of G-MVLG distribution is as the following:
 
:<math>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)}.</math>
 
=== Marginal central moments ===
<math>r^\text{th}</math> marginal central moment of <math>Y_i</math> is as the following:
 
:<math>{\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}.</math>
 
===Marginal expected value and variance===
Marginal expected value <math>Y_i</math> is as the following:
 
:<math> \operatorname{E}(Y_{i})=\frac{1}{\mu_i}\big[\ln(\lambda_i/\delta)+\digamma(\nu)\big], </math>
:<math> \operatorname{var}(Z_i)=\digamma^{[1]}(\nu)/(\mu_i)^2</math>
 
where <math>\digamma(\nu)</math> and <math>\digamma^{[1]}(\nu)</math> are values of [[digamma function|digamma]] and [[trigamma function]]s at <math>\nu</math>, 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 <math>\boldsymbol{T}\sim \mathrm{G}\text{-}\mathrm{MVGB}(\delta,\nu,\boldsymbol{\lambda},\boldsymbol{\mu})</math> is the following:
 
:<math>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}.</math>
 
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==
{{reflist}}
 
{{ProbDistributions|multivariate}}
[[Category:Multivariate continuous distributions]]
[[Category:Probability distributions]]
[[Category:Continuous distributions]]

Latest revision as of 10:08, 27 November 2014

People no longer know on how to have fun while traveling. The results of this study are important because the number one reason kids drop out of organized sports by the time they reach middle school is that they are no longer having fun. If you are bored with the same old routines of exercising and eating right, fulfill your cravings for something different by mixing up your gym routines. fun apps for ipod touch But, that isn't true for communicating with clients and colleagues. Summer jobs are something all college students look forward to. The object of the game is to score as many footballs in the trashcans as possible before time expires. But considering the fact that you have some extra pounds on your body.

Our family entertainment centers have become a popular way for parents to amuse their children with fun and exciting activities, while providing a safe and convenient environment. The object of the game is to toss footballs from one football-filled laundry basket to an empty laundry basket as fast as possible. And today there are so many exercise equipment in the market those are incredibly smart and amazingly easy. Offer to attend as an escort while your ex goes to some event that they love. It is essentially a road trip, usually over 1000 miles that is split over three to seven days. Cows have been sited as a major cause in global warming for their flatulent behavior, and apparently termites are the second largest natural source of methane emissions in the world. The valet took our luggage and we were welcomed by a delightfully warm reception when checking in.

You can pick up a package of water balloons and have a water balloon catch, egg toss or even play Blind Man. It will be like discovering an easy cum speedy way for having instant fun. Kids flash games are a great way to do this, they are easy to build and can easily be made appealing. Putt Putt golf, a huge maze, some shops and of course places to just sit and enjoy the view. Football fans can add to their memorabilia with a custom built number plate complete with the team colours and the badges, plus a slogan or your name. All you have to do is keep the fun games for kids all the time and they will be always willing to play a part. This can be great fun and I have seen a lot of women doing this and really enjoying themselves, but you have to consider the safety issues when selecting this option as a lot of inexperienced riders get injured (including me and I'm experienced.

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Rowing tones your arms and increases your endurance. Take a moment to visit Ned - Rob's - Place to learn more about what a websites for kids should look like. Both countries entered wars in order to defend the other. ' Reduced risk of heart disease, high blood pressure, diabetes and osteoporosis;. What that means is that you cannot legally add an external antenna to the device per F. You are not asked for a sign up amount or a registration fee for bringing the best of movies world into your PC. These also make fun, inexpensive party favors or afternoon craft that kids can make themselves.

Keeping a good balance and monitoring children will help in reaping greater benefits from technology. If you are planning to visit this westernmost country in Europe soon, or have Porteguese roots that you want to learn more about, here are thirteen of the most fascinating facts that are sure to impress anyone about Portugal. We passed by the beautiful gazebo surrounded by a garden of flowers ready to host a wedding ceremony. According to Australia's National Physical Activity Guidelines, the minimum recommended amount of exercise is 30 minutes of 'moderate' activity at least five times a week. The student who guesses the verb correctly gets to act out the next verb. A nearer look shows that the meal is over, dishes cleared and everyone is engaged in the fun of taking part in a board game. Use of video technology for illustrating educational material is much more effective than with the use of books.

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