Complete sufficient statistic (${\mathbb{E}}_{\theta }g(T)=0\forall \theta ⟹g(T)=0\text{ a.s.}$)

A sufficient statistic is complete when every integrable function of it with expectation zero for every parameter is almost surely zero for every parameter. Full regular one-parameter exponential families with natural parameter ranging over an open interval provide a standard completeness result. Completeness makes an unbiased function of a sufficient statistic unique, giving the Lehmann–Scheffé theorem.

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.