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An unbiased estimator that is a function of a complete sufficient statistic is the unique uniformly minimum-variance unbiased estimator, up to almost sure equality. Rao-Blackwellization supplies variance reduction, and completeness supplies uniqueness. The theorem applies to a conditional statistical experiment as well, when unbiasedness, sufficiency and completeness are all evaluated under that conditional family.

Both alternatives can be made explicit. For a conditional bias-corrected normal mean estimate, let $m$ be the observed pooled estimate and $w=n_1/n_2$, with $0<w<1$. Invert its conditional mean:This can be solved by bracketing or by Newton iteration $d_{k+1}=d_k-[g(d_k)-m]/g^{\prime }(d_k)$. Since $r^{\prime }(x)=-r(x)[x+r(x)]$,The variance of a standard normal conditional on exceeding $-x$ is $1-r(x)[x+r(x)]$, strictly between zero and one. Positivity follows from nondegeneracy; the upper bound follows because the conditional mean $r(x)$ exceeds the truncation threshold $-x$. Hence $1-w<g^{\prime }(d)<1$, so the equation has at most one root. As $d\to \infty$, $g(d)\sim d$; as $d\to -\infty$, the truncated stage-1 mean approaches its threshold and $g(d)\sim (1-w)d+w{f}_1/\sqrt{I_1}$, so a root exists for every finite $m$. Equivalently, differentiating the conditional likelihood divides the ordinary likelihood by ${\mathrm{Pr}}_d(C)$ and gives the same score equation. This conditional-likelihood correction is not exactly conditionally unbiased merely because it inverts a mean.

For an illustration, take $I_1=1$, $w=1/2$, $f_1=0$ and $m=0.5$. The equation is $d+\frac{1}{2}\varphi (d)/\Phi (d)=0.5$. Since $g(0)=0.39894$ and $g(0.5)\simeq 0.75458$, the corrected estimate lies between zero and $0.5$; numerical solution gives $\tilde{\delta }\simeq 0.14649$, below the selected ordinary estimate.

For the uniform minimum variance conditionally unbiased estimator, put $I_2=n_2/(2{\sigma }^2)$, $J=I_2-I_1>0$, $A={\hat{\delta }}_1$ and $V={\hat{\delta }}_2^{\ast }$. The fresh estimate $V$ is conditionally unbiased because it is independent of continuation. LetBefore selection, conditional Gaussian calculations give $A\mid T=t\sim N(t,s^2)$. Conditional on $C$ as well, this normal variable is truncated below $c$, soUsing $V=(I_{2}T-I_{1}A)/J$, Rao-Blackwellization therefore givesIts conditional expectation is $\mathbb{E}[V\mid C]=\delta$, and its conditional variance cannot exceed that of $V$. To justify uniform minimum variance, the joint conditional density of $(A,V)$ is a base density on $\{A\ge c\}$ multiplied by $\exp (\delta I_{2}T-{\delta }^2{I}_2/2)/{\mathrm{Pr}}_{\delta }(C)$. Thus $T$ is a complete sufficient statistic in the one-parameter conditional exponential family, whose natural parameter ranges over an open real interval. The Lehmann–Scheffé theorem proves the claim. The orthogonal pooled arm-average statistic is independent of the entire difference process and carries the nuisance common mean. Together with $T$, it gives a complete sufficient statistic in the selected two-parameter normal family, with an open natural-parameter space. Thus allowing that nuisance statistic does not improve the conditional unbiased estimate of the difference. For the same illustration, $I_2=2$, $J=1$ and $s=1/\sqrt{2}$ give ${\hat{\delta }}_{\mathrm{U}}\simeq 0.21102$. It differs from the conditional-likelihood estimate because exact conditional unbiasedness is a different criterion.

A UMVCUE has minimum conditional variance among all estimators unbiased in the selected experiment for every parameter. For independent Gaussian stage estimates $A,V$ with information $I_1,J$, continuation $A\ge c$, total information $I_2=I_1+J$ and pooled estimate $T$, Rao-Blackwellization of the fresh estimate gives $T-(I_{1}s/J)\varphi ((T-c)/s)/\Phi ((T-c)/s)$, where $s^2=1/I_1-1/I_2$. The conditional family has complete sufficient statistic $T$, so the Lehmann–Scheffé theorem establishes the optimum. A conditional-likelihood bias correction does not automatically have this unbiasedness property.