Multivariate gamma function

Template:Short description 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.<ref>Template:Cite journal</ref>

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

<math>

\Gamma_p(a)= \int_{S>0} \exp\left( -{\rm tr}(S)\right)\, \left|S\right|^{a-\frac{p+1}{2}} dS, </math> where <math>|S|</math> denotes the determinant of <math>S</math>. The other one, more useful to obtain a numerical result is:

<math>

\Gamma_p(a)= \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2). </math> In both definitions, <math>a</math> is a complex number whose real part satisfies <math>\Re(a) > (p-1)/2</math>. Note that <math>\Gamma_1(a)</math> reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for <math>p\ge 2</math>:

<math>

\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). </math>

Thus

• <math>\Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2)</math>
• <math>\Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1)</math>

and so on.

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

<math>\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)}</math>

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

The function is derived by Anderson<ref>Template:Cite book</ref> 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 <math>p</math>-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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Derivatives

We may define the multivariate digamma function as

<math>\psi_p(a) = \frac{\partial \log\Gamma_p(a)}{\partial a} = \sum_{i=1}^p \psi(a+(1-i)/2) ,</math>

and the general polygamma function as

<math>\psi_p^{(n)}(a) = \frac{\partial^n \log\Gamma_p(a)}{\partial a^n} = \sum_{i=1}^p \psi^{(n)}(a+(1-i)/2).</math>

Calculation steps

• Since

<math>\Gamma_p(a) = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma\left(a+\frac{1-j}{2}\right),</math>

it follows that

<math>\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).</math>

• By definition of the digamma function, ψ,

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

it follows that

<math>

\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} </math>

References

• 1. Template:Cite journal
• 2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.