Statistic ($T=T(X)$)

A statistic is a measurable function of observed data whose definition does not involve an unknown parameter. It may be scalar or vector valued. Sufficient statistics summarize all information about a parameter in the conditional-distribution sense.

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.