#mean-squared-error

Question 1: Single choice

$X_1, \dots, X_{10}$ are i.i.d. $\textrm{Unif}(0, \theta)$ . Two estimators of $\theta$ are $\hat{\theta} = \max_i X_i$ and $\tilde{\theta} = 2\bar{X}$ . Which statement is correct?

$\hat{\theta}$ is biased low but has the smaller MSE: about $0.015\theta^2$ versus $0.033\theta^2$ 1
$\tilde{\theta}$ is better: it is unbiased, and an unbiased estimator always has the smaller MSE of the two2
$\hat{\theta}$ is better because it is unbiased: on average the largest sample equals $\theta$ 3
Both have MSE $\theta^2/30$ , because they use the same ten samples4

Timed test