Conditional bias-corrected normal mean estimate ($m=d+(w/\sqrt{I_1})\varphi (d\sqrt{I_1}-f)/\Phi (d\sqrt{I_1}-f)$)

In a two-stage normal trial, invert the pooled estimator's conditional mean after continuation to correct its selection shift. The root is unique when the pooled stage-1 weight satisfies $0<w<1$, because its conditional-mean map has derivative between $1-w$ and one. This root also maximizes the corresponding conditional likelihood. It is distinct from exact conditional unbiasedness, which is supplied by a suitable Rao-Blackwell theorem construction.