Consider a collection $X_{1}, \ldots, X_{n}$ of independent random variables with common density function $f(x ; \theta)$ depending on a real parameter $\theta$. What does it mean to say $T$ is a sufficient statistic for $\theta$ ? Prove that if the joint density of $X_{1}, \ldots, X_{n}$ satisfies the factorisation criterion for a statistic $T$, then $T$ is sufficient for $\theta$.

Let each $X_{i}$ be uniformly distributed on $[-\sqrt{\theta}, \sqrt{\theta}]$. Find a two-dimensional sufficient statistic $T=\left(T_{1}, T_{2}\right)$. Using the fact that $\hat{\theta}=3 X_{1}^{2}$ is an unbiased estimator of $\theta$, or otherwise, find an unbiased estimator of $\theta$ which is a function of $T$ and has smaller variance than $\hat{\theta}$. Clearly state any results you use.