Multivariate gamma function

In mathematics, the multivariate gamma function Γ_p 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 $p\times p$ positive-definite real matrices:

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

where $|S|$ denotes the determinant of $S$ . Note that $\Gamma _{1}\left(a\right)$ reduces to the ordinary gamma function. The other one, more useful to obtain a numerical result is:

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

From this, we have the recursive relationships:

\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

and so on.

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

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

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, Mahalabolis etc.

Derivatives

We may define the multivariate digamma function as

\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

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

Calculation steps

Since

\Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma \left(a+{\frac {1-j}{2}}\right),

it follows that

{\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).

By definition of the digamma function, ψ,

{\frac {\partial \Gamma (a+(1-i)/2)}{\partial a}}=\psi (a+(i-1)/2)\Gamma (a+(i-1)/2)

it follows that

{\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}}

References

^ 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.
^ Anderson, T W (1984). An Introduction to Multivariate Statistical Analysis. New York: John Wiley and Sons. pp. Ch. 7. ISBN 0-471-88987-3.

1. James, A. (1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". Annals of Mathematical Statistics. 35 (2): 475–501. doi:10.1214/aoms/1177703550. MR 0181057. Zbl 0121.36605.
2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.

[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.

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