Multivariate gamma function

{{Short description|Multivariate generalization of the 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.{{Cite journal|last=James|first=Alan T.|date=June 1964|title=Distributions of Matrix Variates and Latent Roots Derived from Normal Samples|url=http://projecteuclid.org/euclid.aoms/1177703550|journal=The Annals of Mathematical Statistics|language=en|volume=35|issue=2|pages=475–501|doi=10.1214/aoms/1177703550|issn=0003-4851|doi-access=free}}

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

: $\Gamma_p(a)=
\int_{S>0} \exp\left(
-{\rm tr}(S)\right)\,
\left|S\right|^{a-\frac{p+1}{2}}
dS,$

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

: $\Gamma_p(a)=
\pi^{p(p-1)/4}\prod_{j=1}^p
\Gamma(a+(1-j)/2).$

In both definitions, $a$ is a complex number whose real part satisfies $\Re(a) > (p-1)/2$ . Note that $\Gamma_1(a)$ reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for $p\ge 2$ :

: $\Gamma_p(a) = \pi^{(p-1)/2} \Gamma(a) \Gamma_{p-1}(a-\tfrac{1}{2}) = \pi^{(p-1)/2} \Gamma_{p-1}(a) \Gamma(a+(1-p)/2).$

Thus

$\Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2)$
$\Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1)$

and so on.

This can also be extended to non-integer values of $p$ 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{{Cite book|last=Anderson|first=T W|title=An Introduction to Multivariate Statistical Analysis|publisher=John Wiley and Sons|year=1984|isbn=0-471-88987-3|location=New York|pages=Ch. 7}} 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 $p$ -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.{{cite web|url=https://dlmf.nist.gov/35|title=Chapter 35 Functions of Matrix Argument|work=Digital Library of Mathematical Functions|author=D. St. P. Richards|date=n.d.|access-date=23 May 2022}}

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

:: $\begin{align}
\frac{\partial \Gamma_p(a)}{\partial a} & = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2) \sum_{i=1}^p \psi(a+(1-i)/2) \\[4pt]
& = \Gamma_p(a)\sum_{i=1}^p \psi(a+(1-i)/2).
\end{align}$

References

1. {{cite journal

|title=Distributions of Matrix Variates and Latent Roots Derived from Normal Samples

|last=James |first=A.

|journal=Annals of Mathematical Statistics

|volume=35 |issue=2 |year=1964 |pages=475–501

|doi=10.1214/aoms/1177703550 |mr=181057 | zbl = 0121.36605

|doi-access=free }}

2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.

Category:Gamma and related functions