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 $\Sigma$.
Suppose $H_0 : \mu_1=\mu_2$.
I calculated that $$\hat{\mu_0}=\frac{\sum^{n_1}_{i=1}x_i+\sum^{n_2}_{i=1}y_i}{n_1+n_2}$$
What would $\hat{\Sigma}_0$ equal to?
Would it be equal to
$$\hat{\Sigma_0}=\frac{1}{n_1+n_2}\biggl(\sum^{n_1}_{i=1}(x_i-\hat{\mu_0})(x_1-\hat{\mu_0})^T+\sum^{n_2}_{i=1}(y_i-\hat{\mu_0})(y_i-\hat{\mu_0})^T\biggr)?$$