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.