Independence of MLEs for mean and covariance matrix of multivariate normal distribution

Question

I am looking for information on/the proof of the following theorem. Any help would be great, a book recommendation with the theorem in it would be even better.

Let $X_i \sim \mathcal{N}_N(\mu, \Sigma)$ be independent and identically distributed random vectors for $i\in \{1,...,T\}$. Given the maximum likelihood estimators $\hat{\mu} = \frac{1}{T} \sum_{i=1}^{T} X_i$ and $\hat{\Sigma} = \frac{1}{T} \sum_{i=1}^{T} (X_i - \hat{\mu})(X_i - \hat{\mu})^T$, show that $\hat{\mu}$ and $\hat{\Sigma}$ are independent.

Answer 1

I found the solution in the following book: R. J. Muirhead, Aspects of Multivariate Statistical Theory from 1982. Using Theorem 3.1.2 on page 80, we get that the sample mean and the sample covariance matrix (for all scalars in front of the sum) are independent, which also implies that the sample mean and the MLE sample covariance matrix are independent.

Link to theorem in said book: https://books.google.de/books?id=TKVfbEvrgu4C&lpg=PP1&pg=PA80#v=onepage&q&f=false