Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 4 / 19H / Solution

A sufficient statistic is a function of the sample whose conditional sample distribution, given that statistic, does not depend on the parameter. The factorization criterion for sufficiency states, for a dominated family, that $T$ is sufficient exactly when the joint density has form $g_{\theta }(T(x))h(x)$, where $h$ is independent of $\theta$.

Let $m=X_{(1)}={\min }_i{X}_i$ and $M=X_{(n)}={\max }_i{X}_i$. The joint uniform density isso $\boxed{T=(m,M)}$ is sufficient for $(a,b)$ by the factorization criterion for sufficiency. The uniform order statistics have expectationsFor example, the standardized minimum has survival probability $(1-t{)}^n$ on $[0,1]$, whose integral is $1/(n+1)$; the maximum follows by reflection. Solving these expectation equations gives the unbiased estimatorThe assumption $n\ge 2$ makes the denominator positive.