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The multivariate gamma function (MGF) is an extension of the gamma function for multiple variables. While the gamma function can only handle one input (“x”), the multivariate version can handle many. It is usually defined as:
When m = 1, we drop the m in Γm(a), so the the equation simply becomes the gamma function Γ(a). You may also see the notation Γd(x), although this particular notation also denotes the incomplete regular gamma function in some texts (Gentle, 2007).
Applications of the Multivariate Gamma Function
The MGF has no real practical application but it is used extensively in multivariate statistical analysis. For example, the MGF is included in the probability density function of the Wishart distribution, which is an important part of Bayesian Psychometric Modeling, Signal Processing and many other sub-fields of multivariate analysis. The MGF is also used to derive the general form of the generalized multivariate gamma distribution’s PDF:
Note: A positive definite matrix is a symmetric matrix with all positive eigenvalues. Eigenvalues are special scalars; When you multiply a matrix by a vector and get the same vector as an answer, along with a new scalar, the scalar is called an eigenvalue.
Das & Dey (2007). On Bayesian Inference for Generalized Multivariate Gamma Distribution.
Gentle, J. (2007). Matrix Algebra: Theory, Computations, and Applications in Statistics. Springer Science & Business Media.
Muirhead (2009). Aspects of Multivariate Statistical Theory. John Wiley and Sons.
Wang & Li (2011) “Efficient Gaussian Graphical Model Determination without Approximating Normalizing Constant of the G-Wishart distribution ” http://www.stat.sc.edu/~wang345/RESEARCH/GWishart/GWishart.html
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