2. Let $X_1, \dots, X_n \sim \text{Uniform}(-\theta, \theta)$, $\theta > 0$.
(a) Let $\hat{\theta}_n = X_{(n)} = \max\{X_1, \dots, X_n\}$. Find the bias, se and MSE of this estimator.
Hint: work on $P(\hat{\theta} \le a)$ to figure out its CDF.
(b) Let $\hat{\theta}_n = -X_{(1)} = -\min\{X_1, \dots, X_n\}$. Find the bias, se and MSE of this estimator.
Hint: Figure out the distribution of $-X_i$, and use the results in (a).
(c) Let $\hat{\theta}_n = \max\{|X_1|, \dots, |X_n|\}$. Find the bias, se and MSE of this estimator.
Hint: figure out the distribution of $|X_i|$ by working on $P(|X_i| \le a)$.