Show that the distribution of $(n-1)\overline{X}'\left(S^{-1}-\dfrac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0}\right)\overline{X}$ is $T^2(p-1,n-1)$ where $X_i\sim N_p(\mu,\Sigma)$, where $\Sigma$ is pd but unknown, and it is just known that $\mu=k\mu_0$ for some $k\in\mathbb R$. Also, $nS=\sum_{i=1}^n(X_i-\overline{X})(X_i-\overline{X})'$.

I know a solution where I can view $\mu=k\mu_0$ as $R\mu=0$ for some suitable matrix $R$ orthogonal to $\mu_0$, but I do not want to use that result. I want to directly obtain the result using properties of Wishart and its relation with Hotelling $T^2$, etc.

I know that $nS\sim W_p(\Sigma,n-1)$ and is independent of $\overline{X}$. I will be done if I can somehow identify the middle matrix as inverse of some Wishart with appropriate parameters.

Can anyone give some hint? Thanks.


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