Rao-Blackwell estimator after interim selection ($\tilde{\delta }=S-v\lambda ((c-S)/v)$)

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.