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If $X$ has a Wishart distribution $W_p(n,\Sigma)$ , what's the distribution for $cX$ where $c>0$ ?

I know that for a $\chi^2_n (x)$ distribution with $n$ degrees of freedom, $c\chi^2 $ follows $\Gamma(x;\frac{n}{2},2c)$ for the scaling property of gamma distribution so the answer should not be so obvious.

This question arose because $\frac{1}{n} W$ should be the the correct distribution for the sample covariance, according to what I understand from theory.

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I think I just found an answer but I leave open the question for the sake of the community:

Actually using the theorem found here (1) : $$CXC^t \sim W(n,C\Sigma C^t) $$ and taking $C$ as the diagonal matrix : $$\left( \begin{matrix} \sqrt{c} & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \sqrt{c} \\ \end{matrix} \right) $$

Ergo in this case $cX$ should be $W(n,c\Sigma)$ distributed and then, for the covariance sample purpose should be $\frac{1}{\sqrt{n}}$. In this way $\frac{1}{n}X=\hat{\Sigma}$ should be simply $W(n, \frac{1}{n}\Sigma) $ distributed.

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  • $\begingroup$ You've changed your notation in midstream from $c$ to $n$, it appears. You asked about $cX$, but you are responding with $1/n$... and since $\Sigma$ is a parameter, it doesn't have a distribution (referring to your last sentence.) $\endgroup$ – jbowman Mar 10 '18 at 17:48
  • $\begingroup$ Yes , i know. The question arose for the purpose of covariance sample matrix distrubution, where I have $ c=\frac{1}{n} $ . It's a dummy scalar variable. Of course in this case $X=n\hat{\Sigma}$ has a distribution. I edited the last $\Sigma$ with an hat. $\endgroup$ – R.Lac Mar 10 '18 at 17:50
  • $\begingroup$ Also, which theorem? Please hyperlink or expand on it. $\endgroup$ – Jim Mar 10 '18 at 22:33

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