Let $X_{1}, \ldots, X_{n}$ be i.i.d. $U[0,2 \theta]$ random variables, where $\theta>0$ is unknown.

(a) Derive the maximum likelihood estimator $\hat{\theta}$ of $\theta$.

(b) What is a sufficient statistic? What is a minimal sufficient statistic? Is $\hat{\theta}$ sufficient for $\theta$ ? Is it minimal sufficient? Answer the same questions for the sample mean $\tilde{\theta}:=\sum_{i=1}^{n} X_{i} / n$. Briefly justify your answers.

[You may use any result from the course provided it is stated clearly.]

(c) Show that the mean squared errors of $\hat{\theta}$ and $\tilde{\theta}$ are respectively

$$\frac{2 \theta^{2}}{(n+1)(n+2)} \quad \text { and } \quad \frac{\theta^{2}}{3 n} \text {. }$$

(d) Show that for each $t \in \mathbb{R}, \lim _{n \rightarrow \infty} \mathbb{P}(n(1-\hat{\theta} / \theta) \geqslant t)=h(t)$ for a function $h$ you should specify. Give, with justification, an approximate $1-\alpha$ confidence interval for $\theta$ whose expected length is

$$\left(\frac{n \theta}{n+1}\right)\left(\frac{\log (1 / \alpha)}{n-\log (1 / \alpha)}\right)$$

[Hint: $\lim _{n \rightarrow \infty}\left(1-\frac{t}{n}\right)^{n}=e^{-t}$ for all $t \in \mathbb{R}$.]