 Boundedly complete statistic

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.