# How do I find the “elliptical confidence region” from columns of a matrix that follows the Wishart distribution?

The subject is about the sample mean and the sample covariance estimators and their respective confidence regions for the estimated parameters.

Suppose that $n$ samples are taken from a $p$-variate normal distribution $\mathcal{N}_p(\boldsymbol{\mu},\boldsymbol{\Sigma})$. As far as I know the sample mean estimator has a $p$-variate normal distribution $\mathcal{N}_p(\hat{\boldsymbol{\mu}},\hat{\boldsymbol{\Sigma}}\cdot\frac{1}{n})$ and the sample unbiased covariance estimator has a Wishart distribution $W_p(n-1,\hat{\boldsymbol{\Sigma}}\cdot\frac{1}{n-1})$ with $n-1$ degrees of freedom. The Wishart distribution is a generalization to multiple dimensions of the chi-squared distribution.

The $(1-\alpha)\%$ confidence region for the estimated mean could be obtained as an ellipsoidal inequation:

$\begin{equation}(x-\hat{\boldsymbol{\mu}})^\intercal\left(\displaystyle\frac{\hat{\boldsymbol{\Sigma}}}{n}\right)^{-1}(x-\hat{\boldsymbol{\mu}})\leq \gamma^2\end{equation}$

Where $\gamma^2$ is the value of the Chi-squared CDF for the probability $(1-\alpha)$ and $p$ degrees of freedom. For example, for a confidence region of $95\%$ the value of $\gamma$ is approximately $\sqrt{\chi^2_{_\text{CDF}}(1-\alpha,p)}=\sqrt{5.9915}\approx 2.4477$.

I suppose, based on simulations, that the vectors of the estimated covariance matrix has also an elliptical confidence region. Considering that $\hat{\boldsymbol{\Sigma}}$ is composed by $p$ vectors $[\boldsymbol{w}_1 \boldsymbol{w}_2 \cdots \boldsymbol{w}_p]$ then the confidence region for each vector should be somenting like:

$\begin{equation}(y-\boldsymbol{w}_i)^\intercal \boldsymbol{D}_i^{-1}(y-\boldsymbol{w}_i)\leq \delta_i^2\end{equation}$

The questions that I have are the following:
(1) How do I find the matrices $\boldsymbol{D}_i$ and the parameters $\delta_i$?
(2) The matrix $\boldsymbol{D}_i$ and the parameter $\delta_i$ are the same for all the $\boldsymbol{w}_i$ vectors?