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A sufficient statistic is complete when every integrable function of it with expectation zero for every parameter is almost surely zero for every parameter. Full regular one-parameter exponential families with natural parameter ranging over an open interval provide a standard completeness result. Completeness makes an unbiased function of a sufficient statistic unique, giving the Lehmann–Scheffé theorem.

For the fixed observation interval, $N_3$ has Poisson distribution with mean and variance $3\lambda$. Hence $N_3/3$ is unbiased and has variance $\lambda /3$. Its score function is $N_3/\lambda -3$, and its Fisher information is $3/\lambda$. The Cramér-Rao bound for unbiased estimation of $\lambda$ therefore gives variance at least $\lambda /3$, attained by $N_3/3$.

For the stopping-time observation, $T_{10}$ has Gamma distribution with shape $10$ and rate $\lambda$. Direct integration of its density givesThus $k=9$, andThe score function is $10/\lambda -T_{10}$, so the Fisher information is $10/{\lambda }^2$. Its Cramér-Rao lower bound is ${\lambda }^2/10$, strictly below this estimator's variance.

Nevertheless, $9/T_{10}$ is uniformly minimum-variance unbiased. The observed $T_{10}$ is a sufficient statistic, and its family is complete: if ${\mathbb{E}}_{\lambda }h(T_{10})=0$ for all $\lambda >0$, then ${\int }_0^{\infty }h(t)t^9{e}^{-\lambda t}dt=0$ for all $\lambda >0$. Uniqueness of the Laplace transform, applied after an exponential tilt to make an Lebesgue integrable function, implies $h=0$ almost everywhere. The Lehmann–Scheffé theorem now proves the minimum-variance assertion. Failure to attain the Cramér–Rao bound does not disprove minimum variance within the class of unbiased estimators.

For the actual observations, the two likelihood functions areThey differ by a factor independent of $\lambda$. The likelihood principle consequently requires the same inferential conclusions, since the observed data carry proportional likelihoods. But the two minimum-variance unbiased estimates are $\boxed{10/3\text{ and }3}$ respectively. Minimum-variance unbiased estimation therefore violates the likelihood principle: its unbiasedness and variance refer to the sampling plan, including the stopping rule, rather than solely to the observed Likelihood function.

The Lehmann–Scheffé theorem states that an unbiased estimator which is a function of a complete sufficient statistic is the unique minimum-variance unbiased estimator; equivalently conditioning any unbiased estimator on such a statistic produces it.

Use the unbiased estimator $I=1_{X_1>x}$. Given $T_n=t$, the positive sample vector is uniform on the simplex ${\sum }_i{X}_i=t$: the joint exponential density is constant there, independently of $\lambda$. For $t<x$ the conditional probability is zero. For $t\ge x$, translating the first coordinate by $x$ changes the simplex size from $t$ to $t-x$; its $(n-1)$-dimensional volume ratio is $(1-x/t{)}^{n-1}$. Thusincluding $n=1$, when this is the indicator $1_{T_1\ge x}$ up to a null boundary. Conditional expectation preserves unbiasedness, and completeness from part (i) proves the displayed estimator is the unique minimum-variance unbiased estimator.

It does not attain the Cramér-Rao bound. Equality in that bound requires the centred unbiased estimator to be proportional to the score function, hence affine in $T_n$ at fixed $\lambda$. This estimator is zero on a positive-probability interval and positive elsewhere, so is not affine; the $n=1$ step function is not affine either.

For the last assertion, the remaining observations are independent of the first $k$. Conditional on $T_k$, the internal proportions of those first $k$ are independent of the remaining sum, soNow the tower property for the nested information $\sigma (T_n)\subseteq \sigma (T_k,T_n)$ givesConditioning first on both sums is essential; the sigma-fields generated by $T_k$ and $T_n$ alone are not nested.

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.

An unbiased estimator $T$ is uniformly minimum-variance unbiased if its variance is no larger than that of every other unbiased estimator of the same parameter, for every parameter value. The Lehmann–Scheffé theorem gives such an estimator whenever an unbiased function of a complete sufficient statistic exists. Attaining the Cramér-Rao bound is sufficient for this property under its hypotheses, but is not necessary: for $T\sim \Gamma (n,\lambda )$ with $n>2$, $(n-1)/T$ is unbiased and has variance ${\lambda }^2/(n-2)$, while the information bound is ${\lambda }^2/n$. Completeness follows from uniqueness of the Laplace transform of $h(t)t^{n-1}$, so the estimator is still uniformly optimal among unbiased estimators.

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.