What is the joint distribution $(\hat{\mu_1},\hat{\mu_2},\hat{\Sigma})$?

Let $$X_1,...,X_{n_1}$$ be an i.i.d. sample from $$N_p(\mu_1,\Sigma)$$ and let $$Y_1,...,Y_{n_2}$$ be an independent sample from $$N_p(\mu_2,\Sigma)$$, for some $$\mu_1,\mu_2 \in \mathbb{R}^p$$ and some invertible, $$p\times p$$ positive definite matrix $$\Sigma$$. I found that:

\begin{align} \hat{\mu_1} &= \frac{1}{n_1}\sum^{n_1}_{i=1}x_i, \\ \hat{\mu_2} &= \sum^{n_2}_{i=1}y_i, \\ \hat{\Sigma} &= \frac{1}{n_1+n_2}\biggl(\sum^{n_2}_{i=1}(x_i-\hat{\mu_1})(x_i-\hat{\mu_1})^T+\sum^{n_2}_{i=1}(y_i-\hat{\mu_2})(y_i-\hat{\mu_2})^T\biggr) \end{align}

I'd like to find the joint distribution $$(\hat{\mu_1},\hat{\mu_2},\hat{\Sigma})$$.

I know that $$\hat{\mu_1}\sim N_p(\mu_1,\frac{1}{n_1}\Sigma),\hat{\mu_2}\sim N_p(\mu_2,\frac{1}{n_2}\Sigma)$$

For $$\hat{\Sigma}$$, I'm not quite sure actually. So, I know that for example, if I set $$S_n=\frac{1}{n-1}\sum^n_{i=1}(x_i-\overline{x})(x_i-\overline{x})^T$$ then $$(n-1)S\sim \mathcal{W}_p(\Sigma,n-1)$$. Also, if $$S_m=\frac{1}{m-1}\sum^{m}_{i=1}(x_i-\overline{x})(x_i-\overline{x})^T$$ is independent from $$S_n$$, then $$S_n+S_m\sim \mathcal{W}_p(\Sigma,n+m-2)$$

I think I can then set $$(n_1+n_2)S_x=\frac{n_1+n_2}{n_1+n_2}\bigl(\sum^{n_1}_{i=1}(x_i-\hat{\mu_1})(x_i-\hat{\mu_1})^T\bigr) \sim \mathcal{W}_p(\Sigma,n_1+n_2)$$ and $$S_y=\frac{1}{n_1+n_2}\bigl(\sum^{n_2}_{i=1}(y_i-\hat{\mu_2})(y_i-\hat{\mu_2})^T\bigr)$$, I can then say that $$\hat{\Sigma}=S_x+S_y$$. So by the fact that they are independent, (and since $$S_n+S_m\sim \mathcal{W}_p(\Sigma,n+m-2)$$) I can say that $$(n_1+n_2)\hat{\Sigma}\sim \mathcal{W}_p(\Sigma,n_1+n_2)$$.

Would this be correct?

I also know that $$\hat{\mu_1},\hat{\mu_2},\hat{\Sigma}$$ are all independent. So what exactly would be their joint distribution $$(\hat{\mu_1},\hat{\mu_2},\hat{\Sigma})$$?

• I think you're missing a $\frac{1}{n_2}$. Commented Jan 21, 2021 at 7:47
• Is your problem with finding the (independent) distributions $\hat\mu_1, \hat\mu_2, \hat\Sigma$ or with how to combine them to a joint distribution? Commented Jan 22, 2021 at 17:02
• @SextusEmpiricus yes, how to combine them to a joint distribution
– user255658
Commented Jan 23, 2021 at 8:02

$$\begin{bmatrix} \Sigma_1&0\\0&\Sigma_2 \end{bmatrix}$$