The Bias-Variance Tradeoff

You observe $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$ and want to estimate $\sigma^2$. Consider two estimators: - $\hat{\sigma}^2_{\text{MLE}} = \frac{1}{n} \sum_{i=1}^{n}(X_i - \bar{X})^2$ - $S^2 = \frac{1}{n-1} \sum_{i=1}^{n}(X_i - \bar{X})^2$ (a) Show that $\hat{\sigma}^2_{\text{MLE}}$ is biased. What is its bias? (b) Show that $S^2$ is unbiased. (c) Compute the MSE of both estimators. Which one has lower MSE? (d) Why might you prefer a biased estimator?