Conditional selection bias after futility continuation ($E(S\mid X\ge c)-\delta$)

For independent stage means $X,Y\sim N(\delta ,s^2)$ and combined mean $S=(X+Y)/2$, conditioning on the first-stage sample mean $X$ exceeding $c$ raises that mean. In this equal-stage setting, the estimator bias after continuation is $E(S\mid X\ge c)-\delta =(s/2)\lambda ((c-\delta )/s)$, where $\lambda$ is the upper-tail inverse Mills ratio. It is positive and tends to zero as continuation becomes virtually certain.

Consider independent stage means $X\sim N(\delta ,1/I_1)$ and $Y\sim N(\delta ,1/I_2)$, with positive known information $I_1,I_2$. Continue to stage two only if $X\sqrt{I_1}\ge f$. The pooled mean is $M=wX+(1-w)Y$, where $w=I_1/(I_1+I_2)$. Write $m$ for its observed value. Its conditional mean under a candidate parameter $d$ isHere $\varphi$ and $\Phi$ are the standard normal probability density and cumulative distribution functions. The Conditional bias-corrected normal mean estimate solves $g(d)=m$. This inverts the selection-shifted mean instead of ignoring the fact that the first-stage mean exceeded a threshold.

For the upper-tail inverse Mills ratio $\lambda (a)=\varphi (a)/(1-\Phi (a))$, $g^{\prime }(d)=1-w{\lambda }^{\prime }(f-d\sqrt{I_1})$. The variance of a truncated standard normal variable gives $0<{\lambda }^{\prime }(a)<1$, hence $1-w<g^{\prime }(d)<1$. Since $0<w<1$, $g$ is strictly increasing and ranges over the whole real line; the root exists uniquely. Differentiating the joint normal likelihood conditional on continuation shows that the same root maximizes that conditional likelihood. Inverting a conditional mean does not itself make an estimator conditionally unbiased. The separate Rao-Blackwell estimator after interim selection provides that property in the equal-information setting.

For two independent stage means $X,Y\sim N(\delta ,s^2)$ and continuation $X\ge c$, let $S=(X+Y)/2$ and $v=s/\sqrt{2}$. The second-stage mean is conditionally unbiased. The Rao-Blackwell theorem gives $E(Y\mid S,X\ge c)=S-v\lambda ((c-S)/v)$, using $X\mid S\sim N(S,v^2)$ and truncation. This is a conditionally unbiased estimator using both stages through their combined sample mean.