Completeness and Minimal Sufficiency

A sufficient statistic captures all the information in a sample relevant to a parameter, but there can be many sufficient statistics of varying "size" — the full raw dataset is trivially sufficient, and so is any reversible transformation of it. A minimal sufficient statistic is the most compressed one possible: it's a function of every other sufficient statistic, meaning it throws away every piece of the data that's genuinely irrelevant and nothing more.

Completeness is a separate, less intuitive property: a sufficient statistic $T$ is complete if the only unbiased estimator of zero built from $T$ is the estimator that is identically zero — in other words, there's no way to build a nonzero function of $T$ whose expectation is always zero regardless of the true parameter. Practically, completeness rules out the existence of two different unbiased estimators built from the same sufficient statistic, because if two existed their difference would be a nonzero unbiased estimator of zero.

For example, for $n$ i.i.d. normal observations with unknown mean, the pair (sample mean, sample variance) is a minimal sufficient statistic, and it is also complete — which is exactly the combination the Lehmann-Scheffe theorem needs to certify that functions built from it are the unique best unbiased estimators.

Minimal sufficiency and completeness are logically independent properties in general, but when both hold together for the same statistic, they unlock the strongest results in estimation theory.