I know in general if we have two unbiased estimator $T_1$ and $T_2.$ then $T_1$ is a more efficient estimator of $T_2$ if $Var(T_1)<Var(T_2) $. However in this problem, it asked me to what that an estimator is most efficient:

Problem: Let there be two independent random samples containing $n_1$ and $n_2$ values made out of a random variable $X$ with $\mu$ and $\sigma ^2$.

b) Show that the most efficient estimator $T_1$ can be given in the form $\frac{n_1 \bar{X_1}+n_2 \bar{X_2}}{n_1+n_2}$.

I can do part a), but I am not sure about part b), I do not know what I have to show. I mean if they give me two estimators than I could try to compute variance, but it only gives me one estimator and asks me to show it is the "most efficient". Is there a general way to do problems like this? I would appreciate the help.


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