Lehmann-Scheffé theorem

A statistic $S(𝑿)$ on a random sample of data $𝑿=(X_1,\mathrm{\dots },X_n)$ is said to be a complete statistic if for any Borel measurable function $g$,

$$E(g(S))=0\mathit{\hspace{1em}}\text{implies}\mathit{\hspace{1em}}P(g(S)=0)=1.$$

In other words, $g(S)=0$ almost everywhere whenever the expected value of $g(S)$ is $0$. If $S(𝑿)$ is associated with a family $f(x\mid \theta )$ of probability density functions (or mass function in the discrete case), then completeness of $S$ means that $g(S)=0$ almost everywhere whenever $E_{\theta }(g(S))=0$ for every $\theta$.

Theorem 1 (Lehmann-Scheffé).

If $S\mathit{}\mathrm{(}𝐗\mathrm{)}$ is a complete sufficient statistic and $h\mathit{}\mathrm{(}𝐗\mathrm{)}$ is an unbiased estimator for $\theta$, then, given

$$h_0(s)=E(h(𝑿)|S(𝑿)=s),$$

$h_0(S)=h_0(S(𝑿))$ is a uniformly minimum variance unbiased estimator of $\theta$. Furthermore, $h_{\mathrm{0}}\mathit{}\mathrm{(}S\mathrm{)}$ is unique almost everywhere for every $\theta$.