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Multivariate gamma function - Wikipedia

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In mathematics, the multivariate gamma function Γp is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate beta distribution.[1]

It has two equivalent definitions. One is given as the following integral over the {\displaystyle p\times p} positive-definite real matrices:

{\displaystyle \Gamma _{p}(a)=\int _{S>0}\exp \left(-{\rm {tr}}(S)\right)\,\left|S\right|^{a-{\frac {p+1}{2}}}dS,}

where {\displaystyle |S|} denotes the determinant of {\displaystyle S}. The other one, more useful to obtain a numerical result is:

{\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+(1-j)/2).}

In both definitions, {\displaystyle a} is a complex number whose real part satisfies {\displaystyle \Re (a)>(p-1)/2}. Note that {\displaystyle \Gamma _{1}(a)} reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for {\displaystyle p\geq 2}:

{\displaystyle \Gamma _{p}(a)=\pi ^{(p-1)/2}\Gamma (a)\Gamma _{p-1}(a-{\tfrac {1}{2}})=\pi ^{(p-1)/2}\Gamma _{p-1}(a)\Gamma (a+(1-p)/2).}

Thus

  • {\displaystyle \Gamma _{2}(a)=\pi ^{1/2}\Gamma (a)\Gamma (a-1/2)}
  • {\displaystyle \Gamma _{3}(a)=\pi ^{3/2}\Gamma (a)\Gamma (a-1/2)\Gamma (a-1)}

and so on.

This can also be extended to non-integer values of {\displaystyle p} with the expression:

{\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}{\frac {G(a+{\frac {1}{2}})G(a+1)}{G(a+{\frac {1-p}{2}})G(a+1-{\frac {p}{2}})}}}

Where G is the Barnes G-function, the indefinite product of the Gamma function.

The function is derived by Anderson[2] from first principles who also cites earlier work by Wishart, Mahalanobis and others.

There also exists a version of the multivariate gamma function which instead of a single complex number takes a {\displaystyle p}-dimensional vector of complex numbers as its argument. It generalizes the above defined multivariate gamma function insofar as the latter is obtained by a particular choice of multivariate argument of the former.[3]

We may define the multivariate digamma function as

{\displaystyle \psi _{p}(a)={\frac {\partial \log \Gamma _{p}(a)}{\partial a}}=\sum _{i=1}^{p}\psi (a+(1-i)/2),}

and the general polygamma function as

{\displaystyle \psi _{p}^{(n)}(a)={\frac {\partial ^{n}\log \Gamma _{p}(a)}{\partial a^{n}}}=\sum _{i=1}^{p}\psi ^{(n)}(a+(1-i)/2).}
  • Since
{\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma \left(a+{\frac {1-j}{2}}\right),}
it follows that
{\displaystyle {\frac {\partial \Gamma _{p}(a)}{\partial a}}=\pi ^{p(p-1)/4}\sum _{i=1}^{p}{\frac {\partial \Gamma \left(a+{\frac {1-i}{2}}\right)}{\partial a}}\prod _{j=1,j\neq i}^{p}\Gamma \left(a+{\frac {1-j}{2}}\right).}
{\displaystyle {\frac {\partial \Gamma (a+(1-i)/2)}{\partial a}}=\psi (a+(i-1)/2)\Gamma (a+(i-1)/2)}
it follows that
{\displaystyle {\begin{aligned}{\frac {\partial \Gamma _{p}(a)}{\partial a}}&=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+(1-j)/2)\sum _{i=1}^{p}\psi (a+(1-i)/2)\\[4pt]&=\Gamma _{p}(a)\sum _{i=1}^{p}\psi (a+(1-i)/2).\end{aligned}}}
  1. ^ James, Alan T. (June 1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". The Annals of Mathematical Statistics. 35 (2): 475–501. doi:10.1214/aoms/1177703550. ISSN 0003-4851.
  2. ^ Anderson, T W (1984). An Introduction to Multivariate Statistical Analysis. New York: John Wiley and Sons. pp. Ch. 7. ISBN 0-471-88987-3.
  3. ^ D. St. P. Richards (n.d.). "Chapter 35 Functions of Matrix Argument". Digital Library of Mathematical Functions. Retrieved 23 May 2022.