Question: If \({{\hat{\sigma }}^{2}}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{\left( {{X}_{i}}-\bar{X} \right)}^{2}}}\), show that \(E\left( {{{\hat{\sigma }}}^{2}} \right)=\left( \frac{n-1}{n} \right)\sigma _{X}^{2}\) and that \(\operatorname{var}\left( {{{\hat{\sigma }}}^{2}} \right)=\frac{2\left( n-1 \right)\sigma _{X}^{4}}{{{n}^{2}}}\).

Use these two results to find \(MSE\left( {{{\hat{\sigma }}}^{2}} \right)\). (Hint: Use the fact that \(\left( \frac{n-1}{n} \right)s_{X}^{2}\). Does \({{\hat{\sigma }}^{2}}\) have a smaller mean squared error than does s_X²? (Justify your answer.)