2009-77.md

course	course_year	question_number	tags	title	year
Statistics	IB	77	IB 2009 Statistics	Paper 4, Section II, H	2009

What is a sufficient statistic? State the factorization criterion for a statistic to be sufficient.

Suppose that $X_{1}, \ldots, X_{n}$ are independent random variables uniformly distributed over $[a, b]$, where the parameters $a<b$ are not known, and $n \geqslant 2$. Find a sufficient statistic for the parameter $\theta \equiv(a, b)$ based on the sample $X_{1}, \ldots, X_{n}$. Based on your sufficient statistic, derive an unbiased estimator of $\theta$.