course |
course_year |
question_number |
tags |
title |
year |
Statistics |
IB |
74 |
|
Paper 1, Section II, H |
2011 |
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 $-n \log \theta$.