3
$\begingroup$

Matrix gamma distribution (defined for example in http://en.wikipedia.org/wiki/Matrix_gamma_distribution) is one way to generalize Wishart distribution. In our course work that distribution was used to define generalized matrix t-distribution as a continuous mixture of Gaussians (http://en.wikipedia.org/wiki/Matrix_t-distribution#Generalized_matrix_t-distribution). I got interested in this distribution and I found it useful also in my thesis.

The question: what is the expectation and the variance (componentwise, Var(X_{ij})) of Matrix gamma distribution?

Improve this question
$\endgroup$

2 Answers 2

Reset to default
2
$\begingroup$

The matrix Gamma distribution (as defined in Wikipedia) $MG_p(\alpha,\beta,\Sigma)$ is nothing but the Wishart distribution $W_p\left(2\alpha, \frac{\beta}{2}\Sigma\right)$. The moments and the variances of the Wishart distribution are well-known.

Improve this answer
$\endgroup$
2
$\begingroup$

A good source on this topic is chapter 7 of Roob Muirhead's book, "Aspects of Multivariate Statistical Theory", or chapter 3 of Peter Forrester's "Log-Gases and Random Matrices". In particular, there are some special functions called zonal polynomials which satisfy $$ \int e^{-{\rm tr}(XZ)}\det(X)^{a-(N+1)/2}P_\lambda(X)dX\propto \det(Z)^{-a}P_\lambda(Z^{-1}).$$

This holds for all classes of matrices, i.e. real, complex or quaternion. From this you can get all kinds of statistics (expectation, variance, etc).

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .