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

A statistic is complete for a parameter family if the displayed implication holds for every measurable $g$ integrable under all parameter values. This property does not itself assert sufficiency; a complete sufficient statistic has both properties. A nonconstant unbiased estimator of a known constant provides a direct obstruction to completeness.

Bounded completeness requires the defining implication for a complete statistic only for bounded measurable functions. Completeness implies it, but the converse need not hold. If a sufficient statistic $T$ gives a parameter-independent event probability $c$ and the conditional probability $S(T)$ of that event is nonconstant, then $S-c$ is a bounded zero-mean obstruction to bounded completeness.