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**lllll** Suppose that $\displaystyle Y_1, \ Y_2, \ \dotso,\ Y_n$ is a random sample from a normal distribution with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$. Given two unbiased estimators, find their relative efficiency.

$\displaystyle \sigma^2_1 = S^2 = \frac{1}{n-1} \sum^n_{i=1} (Y_i-\overline{Y})^2$ and $\displaystyle \sigma^2_2 =\frac{1}{2}(Y_1-Y_2)^2$

by definition the relative efficiency is $\displaystyle \frac{V[Y_2]}{V[Y_1]}$ so in this case I should have:

$\displaystyle \frac{\sigma^2_2}{\sigma^2_1} = \frac{\frac{1}{2}(Y_1-Y_2)^2}{\frac{1}{n-1} \sum^n_{i=1} (Y_i-\overline{Y})^2}$

I was thinking of expanding it out but all I got was $\displaystyle \frac{\sigma^2_2}{\sigma^2_1}$.