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 [math]\displaystyle{ p \times p }[/math] positive-definite real matrices:

[math]\displaystyle{ \Gamma_p(a)= \int_{S\gt 0} \exp\left( -{\rm tr}(S)\right)\, \left|S\right|^{a-\frac{p+1}{2}} dS, }[/math]

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

[math]\displaystyle{ \Gamma_p(a)= \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2). }[/math]

In both definitions, [math]\displaystyle{ a }[/math] is a complex number whose real part satisfies [math]\displaystyle{ \Re(a) \gt (p-1)/2 }[/math]. Note that [math]\displaystyle{ \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]\displaystyle{ p\ge 2 }[/math]:

[math]\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). }[/math]

Thus

• [math]\displaystyle{ \Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2) }[/math]
• [math]\displaystyle{ \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]\displaystyle{ p }[/math] with the expression:

[math]\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)} }[/math]

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 [math]\displaystyle{ 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.^[3]

Derivatives

We may define the multivariate digamma function as

[math]\displaystyle{ \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]\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). }[/math]

Calculation steps

• Since

[math]\displaystyle{ \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]\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). }[/math]

• By definition of the digamma function, ψ,

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

it follows that

[math]\displaystyle{ \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. 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. 
• 2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.

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