# How can i test equality of means of two normal populations when $\Sigma$ is known and unknown?

Let's say $$\{x_i\}_{i=1}^m,\{y_i\}_{j=1}^n$$ are i.i.d samples from two independent multivariate normal populations $$N_d(\mu_1,\Sigma)$$ and $$N_d(\mu_2,\Sigma)$$. How can I run a hypothesis test to test $$H_0:\mu_1=\mu_2$$ vs $$H_1:\mu_1\neq \mu_2$$ when (a) $$\Sigma$$ is known and (b) $$\Sigma$$ is unknown?

(a) I use Hotelling $$T^2$$ statistic such that $$T^2 = n(\bar{x}-\mu_0)^T\Sigma_0^{-1}(\bar{x}-\mu_0),$$ which is $$\chi^2_p$$ distributed.

Is this correct? Then, for when $$\Sigma$$ is UNKNOWN, would we just use the MLE of $$\Sigma$$ and use $$T^2$$ as before? Or would we use an F-statistic? I don't really understand.

The independence and multivariate normality assumptions immediately imply $$\begin{pmatrix} x_1 \\ \vdots \\ x_m \\ y_1 \\ \vdots \\ y_n \end{pmatrix} \sim \mathcal{N}_{\left(m+n\right)d} \left( \begin{pmatrix} \mathbf{1}_m \otimes \mu_1\\ \mathbf{1}_n \otimes \mu_2 \end{pmatrix} , I_{m+n} \otimes \Sigma \right).$$ Hence, $$\bar{x} - \bar{y} \sim \mathcal{N}_d \left(\mu_1 - \mu_2, \frac{m+n}{mn} \cdot \Sigma \right)$$ and $$\frac{mn}{m+n} \cdot \left(\bar{x} - \bar{y}\right)^\top \Sigma^{-1} \left(\bar{x} - \bar{y}\right) \mathrel{=:} T_a \overset{H_0}{\sim} \chi^2\left(d\right).$$
If $$\Sigma$$ is unknown we can start from $$m \cdot S_x + n \cdot S_y \sim W_d\left(\Sigma, m + n -2 \right),\\ S_x=\frac{1}{m}\sum_{i=1}^{m}\left(x_i - \bar x\right)\left(x_i -\bar x\right)^\top, \\ S_y=\frac{1}{n}\sum_{i=1}^{n}\left(y_i - \bar y\right)\left(y_i -\bar y\right)^\top,$$ where $$W_d\left(\Sigma, m + n -2\right)$$ denotes the Wishart distribution with scale matrix $$\Sigma \in \mathbb{R}^{d \times d}$$ and $$m + n -2$$ degrees of freedom.
Since $$S = \left(m \cdot S_x + n \cdot S_y\right)/\left(m + n\right)$$ is independent of $$\left(\bar{x} - \bar{y}\right)$$, we get $$\frac{mn\left(m + n - 2\right)}{\left(m + n \right)^2} \cdot \left(\bar{x} - \bar{y}\right)^\top S^{-1} \left(\bar{x} - \bar{y}\right) \mathrel{=:} T_b \overset{H_0}{\sim} T^2\left(d, m + n - 2\right),$$ where $$T^2\left(d, m + n - 2\right)$$ denotes the Hotelling $$T$$-squared distribution with $$d$$ and $$m + n - 2$$ degrees of freedom. Note that $$S$$ is the MLE for $$\Sigma$$ (see, e.g., this derivation which can readily be extended to our setting).
The connection to the $$F$$-distribution is given by $$T_b \sim T^2\left(d, m + n - 2\right) \iff \frac{m + n -d - 1}{d\left(m + n - 2\right)} \cdot T_b \mathrel{=:} T_c \sim F\left(d, m + n -d -1\right),$$ where $$F\left(d, m + n -d -1\right)$$ denotes the $$F$$-distribution with $$d$$ and $$m + n -d -1$$ degrees of freedom.
With that you can proceed as usual and compare the realized value of $$T_a$$ or $$T_c$$ to the $$\left(1-\alpha\right)$$-quantile of the $$\chi^2\left(d\right)$$ or $$F\left(d, m + n -d -1\right)$$ distribution, respectively, to carry out a hypothesis test at a $$\alpha \cdot 100\%$$ significance level.