The Lehmann-Scheffe Theorem

Suppose you want to estimate some unknown quantity — a mean daily return, say — and you have several unbiased estimators to choose from. Which one is genuinely "best," in the sense of having the smallest possible variance among every unbiased estimator anyone could ever construct, not just the ones you happened to try? Checking every possible unbiased estimator one by one is impossible; there are infinitely many. The Lehmann-Scheffe theorem gives a shortcut: find a statistic that is both complete and sufficient for the parameter, and any unbiased function of it is automatically the unique minimum-variance unbiased estimator (UMVUE) — no further search needed.

The intuition is that a complete sufficient statistic squeezes out every last drop of relevant information in the data with no redundancy left over, so there's exactly one way to build an unbiased estimator from it. Completeness rules out the possibility of two different unbiased functions of the same sufficient statistic both existing, which is what makes the estimator unique rather than merely "good."

For example, with $n$ i.i.d. normal returns, the sample mean $\bar X$ is a function of the complete sufficient statistic for the population mean $\mu$, and $\bar X$ is unbiased for $\mu$ — so Lehmann-Scheffe says $\bar X$ is the UMVUE. No cleverer combination of the data can produce a lower-variance unbiased estimate of the mean.

The theorem is a certificate of optimality, not a recipe for finding the sufficient statistic itself — that step (usually via the factorization theorem, then checking completeness) is separate work.