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.