I have the following problem.
$X_i \overset{IID}{\sim} Normal(\mu, \sigma_1^2) $, $Y_j \overset{IID}{\sim} Normal(\mu, \sigma_2^2), i = 1, \cdots, m, j=1, \cdots n $
Find the MLE for the $\hat{\mu}^{MLE}$, where both $\sigma_1^2, \sigma_2^2$ are unknown.
I bumped into the following old paper : Estimating the Common Mean of Several Normal Populations.
I understand they suggest
$$ \hat{\mu}^{MLE} = \frac {m \bar{x}/s_1^2 + n \bar{y}/s_2^2} {1/s_1^2 + 1/s_2^2} $$
where $s_1^2 = \frac{\sum_{i=1}^m (x_i - \bar{x})^2}{m-1}, s_2^2 = \frac{\sum_{j=1}^n (y_j - \bar{y})^2}{n-1}$, the sample variance of each.
I understand $\mathbb{E}(\hat{\mu}^{MLE}) = \mu$, because sample mean and sample variance is are independent in IID normals, we have
$$ \mathbb{E}(\hat{\mu}^{MLE}) = \mathbb{E}\left[ \mathbb{E}(\hat{\mu}^{MLE}| s_1^2, s_2^2) \right] = \mathbb{E}\left[ \mu \right] = \mu $$
But the variance of $\hat{\mu}^{MLE}$ was not suggested.
My attempt $$ \mathrm{Var}(\hat{\mu}^{MLE}) = \mathbb{E}(\mathrm{Var}(\hat{\mu}^{MLE}|s_1^2, s_2^2)) + \mathrm{Var} (\mathbb{E}(\hat{\mu}^{MLE}|s_1^2, s_2^2))) $$
The second term is zero, because conditional mean yields constant. But the mean of variance part, I have variance
$$ \mathrm{Var}(\hat{\mu}^{MLE}|s_1^2, s_2^2) = A^2 m \sigma_1^2 + (1-A)^2 n \sigma_2^2 $$
where $A = \frac {1/s_1^2} {1/s_1^2 + 1/s_2^2}$.
But I can't proceed. Is there anyone to help me with algebra/Any papers to refer to?
