Chapter 9.5, Problem 57E

Refer to Exercise 9.18. Is the estimator of σ² given there an MVUE of σ²?

9.18 In Exercise 9.17, suppose that the populations are normally distributed with ${\sigma }_1^2={\sigma }_2^2={\sigma }^2.$ Show that

$\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(X_i-\bar{X}\right)}^2}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(Y_i-\bar{Y}\right)}^2}}{2n-2}$

is a consistent estimator of σ².