Is there a name for the distribution resulting from the sum of outer products of t-distributed random vectors?

Alternatively, is there a matrix-valued distribution with the support of positive definite matrices that exhibits a higher degree of tail dependence than the Wishart distribution?


Sutradhar and Ali (1989) - A Generalization of the Wishart Distribution for the Elliptical Model and Its Moments for the Multivariate t Model

They show:

Let the p-dimensional random vectors $X_1, \ldots, X_n$ be i.i.d. having a multivariate t-distribution given by $$ f\left(x_1,x_2,\ldots,x_N\right) = K\left(N,p\right) |\Lambda|^{-N/2} \{\nu + \sum_{j=1}^N \left(x_j - \theta\right)' \Lambda^{-1} \left(x_j - \theta\right) \}^{-(\nu + Np)/2}, $$ where $\nu>0$ and $K\left(N,p\right) = \nu^{\nu/2} \Gamma\{\left(\nu + N p \right)/2\}/\{\pi^{Np/2} \Gamma\left(\nu/2\right)\} $.

Then the $p \times p$ matrix $A = \sum_{j=1}^N \left(X_j - \bar{X}\right) \left(X_j - \bar{X}\right)'$ has the density $$ g\left(A\right) = c\left(n,p;\nu\right)|\Lambda|^{-n/2} |A|^{(n-p-1)/2}\{\nu + tr\left(\Lambda^{-1} A \right)\}^{-(\nu+n p)/2}, $$ for $A>0, \Lambda >0, n \geq p$ where $n=N-1$ and $$ c\left(n,p;\nu\right) = \frac{\nu^{\nu/2}\Gamma\{\left(\nu+n p\right)/2\}}{\Gamma\left(\nu/2\right) \Gamma_p\left(n/2\right)}. $$

They also provide moments for this "t-Wishart" distribution.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.