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Let $t_1$ be the most efficient estimator and $t_2$ be the less efficient estimator with efficiency $e$ and let $r$ be correlation coefficient between the two estimator $t_1$ and $t_2$.Define relationship between $e$ and $r$.

So ofcourse $V(t_1)<V(t_2)$

Now I am not sure if $e=\dfrac{V(t_1)}{V(t_2)}$ or $e=\dfrac{V(t_2)}{V(t_1)}$ because in question it does not say relative efficiency with respect to $t_1$ or $t_2$.

I tried with both of them taking $e=\dfrac{V(t_1)}{V(t_2)}$ for now

$r=\dfrac{COV(t_1,t_2)}{\sqrt{V(t_1)V(t_2)}}$

$ \ \ =\dfrac{E(t_1t_2)-E(t_1)Et_2)}{{{eV(t_2)}}}$

I am not sure how to proceed now .

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  • $\begingroup$ There is a relation if you are looking at the class of unbiased estimators of some function of $\theta$, your parameter of interest. $\endgroup$ – StubbornAtom Jan 11 at 18:30

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