Let $X_{1}, \ldots, X_{n}$ be independent random variables with probability mass function $f(x ; \theta)$, where $\theta$ is an unknown parameter.

(i) What does it mean to say that $T$ is a sufficient statistic for $\theta$ ? State, but do not prove, the factorisation criterion for sufficiency.

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

Now consider the case where $f(x ; \theta)=\frac{1}{x !}(-\log \theta)^{x} \theta$ for non-negative integer $x$ and $0<\theta<1$.

(iii) Find a one-dimensional sufficient statistic $T$ for $\theta$.

(iv) Show that $\tilde{\theta}=\mathbb{\prod}_{\left\{X_{1}=0\right\}}$ is an unbiased estimator of $\theta$.

(v) Find another unbiased estimator $\widehat{\theta}$which is a function of the sufficient statistic $T$ and that has smaller variance than $\tilde{\theta}$. You may use the following fact without proof: $X_{1}+\cdots+X_{n}$ has the Poisson distribution with parameter $-n \log \theta$.