(a) What does it mean to say a statistic $T$ is sufficient for an unknown parameter $\theta$ ? State the factorisation criterion for sufficiency and prove it in the discrete case.

(b) State and prove the Rao-Blackwell theorem.

(c) Let $X_{1}, \ldots, X_{n}$ be independent samples from the uniform distribution on $[-\theta, \theta]$ for an unknown positive parameter $\theta$. Consider the two-dimensional statistic

$$T=\left(\min _{i} X_{i}, \max _{i} X_{i}\right) .$$

Prove that $T$ is sufficient for $\theta$. Determine, with proof, whether or not $T$ is minimally sufficient.