$\newcommand{\bx}{\mathbf{x}}$ $\newcommand{\bSigma}{\boldsymbol{\Sigma}}$ $\newcommand{\bE}{\mathbf{E}}$ $\newcommand{\bD}{\mathbf{D}}$

Consider the multivariate central t distribution with p.d.f. \begin{align} f(\bx| \nu, \bSigma) = \frac{\nu^{\frac{\nu}{2}}\Gamma(\frac{\nu+p}{2})}{\pi^\frac{p}{2}\Gamma(\frac{\nu}{2})} |\bSigma|^{-\frac{1}{2}} \left(\nu + \bx'\bSigma^{-1}\bx\right)^{-\frac{\nu + p}{2}}, \end{align} where $\nu$ is a scalar parameter and $\bSigma$ is a $p$ by $p$ symmetric positive definite parameter matrix.

The log-likelihood function is $$ \mathcal{L(\bSigma, \nu| \bx)} = c(\nu) -\frac{1}{2} log(\left|\bSigma\right|) - \frac{\nu + p}{2} log(\nu + \bx'\bSigma^{-1}\bx). $$

Does anyone know the Fisher information matrix w.r.t. $\bSigma$, that is $- \bE \left[ \frac{\partial^2 \mathcal{L}}{\partial vech(\bSigma) \partial vech(\bSigma)'}\right]$ or $\bE \left[ \frac{\partial \mathcal{L}}{\partial vech(\bSigma)} \left(\frac{\partial \mathcal{L}}{\partial vech(\bSigma)}\right)' \right]$, or where I could find it?


1 Answer 1


The full derivation is given in the appendix of the classic JASA artice "Robust statistical modelling using the t-distribution" by Lange et al (1989), which is available for free here.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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