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

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?

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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.

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