Chapter 9.3, Problem 16E

Refer to Exercise 9.5. Is ${\hat{\sigma }}_2^2$ a consistent estimator of σ²?

9.5 Suppose that Y₁, Y₂,..., Y_n is a random sample from a normal distribution with mean µ and variance σ². Two unbiased estimators of σ² are

${\hat{\sigma }}_1^2=S^2=\frac{1}{n-1}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(Y_i-\bar{Y}\right)}^2\text{ and }}{\hat{\sigma }}_2^2=\frac{1}{2}{\left(Y_1-Y_2\right)}^2.$

Find the efficiency of ${\hat{\sigma }}_1^2$ relative to ${\hat{\sigma }}_2^2$.