= Basu's theorem
{c}
= Basu theorem
{c}
{synonym}
A <complete sufficient statistic> is independent of every <ancillary statistic>. To prove this, let $S$ be complete and sufficient, let $A$ be ancillary, and take any bounded measurable function $h$. By sufficiency, $g(S)=E_\theta[h(A)\mid S]$ has a parameter-independent version; by ancillarity, $c=E_\theta h(A)$ is independent of $\theta$. Thus $E_\theta[g(S)-c]=0$ for every $\theta$. Completeness gives $g(S)=c$ almost surely. Choosing $h$ to be the indicator of each measurable event in the ancillary space proves <independence>. Completeness is essential: a sufficient statistic may itself contain a nonconstant ancillary, as in the minimum and maximum of a uniform location sample.
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