Let $\{f(\cdot \mid \theta): \theta \in \Theta\}$ be a parametric family of densities for observation $X$. What does it mean to say that the statistic $T \equiv T(X)$ is sufficient for $\theta$ ? What does it mean to say that $T$ is minimal sufficient?

State the Rao-Blackwell theorem. State the Cramér-Rao lower bound for the variance of an unbiased estimator of a (scalar) parameter, taking care to specify any assumptions needed.

Let $X_{1}, \ldots, X_{n}$ be a sample from a $U(0, \theta)$ distribution, where the positive parameter $\theta$ is unknown. Find a minimal sufficient statistic $T$ for $\theta$. If $h(T)$ is an unbiased estimator for $\theta$, find the form of $h$, and deduce that this estimator is minimum-variance unbiased. Would it be possible to reach this conclusion using the Cramér-Rao lower bound?