Estimating the average of two gaussians' mean

Question

Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$.

I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$.

In my setting, $\sigma_1,\sigma_2$ are known and we want to estimate the average of the means (e.g., what is the MLE of it?).

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In the special case where we know that $\mu=\mu_1=\mu_2$ (i.e., we estimate the same quantity from observations with different variances), it is known that the MLE is:

$$ X\cdot \frac{\sigma_2^2}{\sigma_1^2+\sigma_2^2} + Y\cdot \frac{\sigma_1^2}{\sigma_1^2+\sigma_2^2}. $$

How does this generalize to arbitrary $\mu_1,\mu_2$?

Answer 1

The maximum likelihood estimator (MLE) for $(\mu_1,\mu_2)$ is $(X,Y)$. So, by the functional invariance of the MLE (that is, simply by definition), the MLE of $g(\mu_1,\mu_2):=(\mu_1+\mu_2)/2$ is $g(X,Y):=(X+Y)/2$, which also, obviously, maximizes the profile likelihood $$L_{X,Y}(\mu):=\sup\{L_{X,Y}(\mu_1,\mu_2)\colon(\mu_1+\mu_2)/2=\mu\}$$ in $\mu$, where $L_{X,Y}(\mu_1,\mu_2)$ is the likelihood.

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One may note that the MLE $(X+Y)/2$ of $(\mu_1+\mu_2)/2$ does not depend on $(\sigma_1,\sigma_2)$.