# MLE for $\Sigma$ for MVN

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'd like to find the likelihood function $$L(\mu_1,\mu_2,\Sigma)$$ of the commbined sample:

In my book, the likelihood function of $$X_1,...,X_n \sim N_p(\mu,\Sigma)$$ is given by

$$\frac{1}{(2\pi)^{np/2}\text{det}(\Sigma)^{n/2}}\exp\biggl(-1/2\bigl(\sum^n_{i=1}(x_i-\mu)^T\Sigma^{-1}(x_i-\mu)\bigr)\biggr)$$

So, $$L(\mu_1,\mu_2,\Sigma)=$$ $$\frac{1}{(2\pi)^{p(n_1+n_2)/2}\text{det}(\Sigma)^{\frac{n_1+n_2}{2}}}\exp\biggl(-1/2\sum^{n_1}_{i=1}(x_i-\mu_1)^T\Sigma^{-1}(x_i-\mu_1)-1/2\sum^{n_2}_{i=1}(y_i-\mu_2)^T\Sigma^{-1}(y_i-\mu_2)\biggr)$$

So I would like to find the MLE of $$\Sigma$$:

$$\mathcal{L}(\mu_1,\mu_2,\Sigma)\propto\det(\Sigma)^{\frac{n_1+n_2}{2}}\exp\biggl(-\frac{1}{2}\bigl(\sum^{n_1}_{i=1}(x_i-\mu_1)^T\Sigma^{-1}(x_i-\mu_1)+\sum^{n_2}_{j=1}(y_j-\mu_2)^T\Sigma^{-1}(y_j-\mu_2)\bigr)\biggr)$$

We can say that $$(x_i-\mu_1)^T\Sigma^{-1}(x_i-\mu_1)+(y_j-\mu_2)^T\Sigma^{-1}(y_j-\mu_2)$$ is the trace of a $$1\times 1$$ matrix, so

$$=\det(\Sigma)^{\frac{n_1+n_2}{2}}\exp\biggl(-\frac{1}{2}\bigl(\sum^{n_1}_{i=1}Tr((x_i-\mu_1)(x_i-\mu_1)^T\Sigma^{-1})+\sum^{n_2}_{j=1}Tr((y_j-\mu_2)(y_j-\mu_2)^T\Sigma^{-1}\bigr)\biggr)$$

$$=\det(\Sigma)^{\frac{n_1+n_2}{2}}\exp\biggl(-\frac{1}{2}\bigl(Tr(S_x\Sigma^{-1})+Tr(S_y\Sigma^{-1})\bigr)\biggr)$$

where $$S_x=\sum^{n_1}_{i=1}(x_i-\mu_1)(x_i-\mu_1)^T\in \mathbb{R}^{p\times p}$$

$$S_y=\sum^{n_2}_{j=1}(y_j-\mu_2)(y_j-\mu_2)^T\in \mathbb{R}^{p\times p}$$

and with $$S=S^{1/2}S^{1/2}$$:

$$=\det(\Sigma)^{\frac{n_1+n_2}{2}}\exp(-\frac{1}{2}\biggl(Tr\bigl(S^{1/2}_x\Sigma^{-1}S^{1/2}_x\bigr)+Tr\bigl(S^{1/2}_y\Sigma^{-1}S_y^{1/2}\bigr)\biggr)$$

Let $$B_x=S_x^{1/2}\Sigma^{-1}S_x^{1/2},B_y=S_y\Sigma^{-1}S_y^{1/2}$$

So,

$$=\det(S_x)^{\frac{-n_1}{2}}\det(B_x)^{\frac{n_1}{2}}\det(S_y)^{\frac{-n_2}{2}}\det(B_y)^{\frac{n_2}{2}}\exp\biggl(\frac{-1}{2}Tr(B_x+B_y)\biggr)$$

and now i'm stuck. What i'm trying to do is on this wikipedia page, but i'm struggling to use it on my example. Any help would be highly appreciated.

$$S^*=S_x+S_y$$ should take the role of $$S$$ in the Wikipedia proof. Use $$Tr(S_x\Sigma^{-1})+Tr(S_y\Sigma^{-1})=Tr(S^* \Sigma^{-1})$$, then you should be able to go on as on Wikipedia.