Bias-corrected bootstrap interval

Assume AC, as required by the standard normal law. Input observed estimate $\widehat{\theta }$, a nonempty list of $B$ bootstrap estimates ${\widehat{\theta }}_1^{\ast },\dots ,{\widehat{\theta }}_B^{\ast }$ generated under IID empirical bootstrap sample and mean statistic, and $0<\alpha <1$. Use the explicit midrank tie convention ${\widehat{p}}_0=\frac{\mathrm{\#}\{b:{\widehat{\theta }}_b^{\ast }<\widehat{\theta }\}+\frac{1}{2}\mathrm{\#}\{b:{\widehat{\theta }}_b^{\ast }=\widehat{\theta }\}}{B}\mathrm{.}$ Let $z_0={\mathrm{\Phi }}^{-1}({\widehat{p}}_0)$, where $\mathrm{\Phi }$ is the standard normal CDF. Use the extended endpoint convention ${\mathrm{\Phi }}^{-1}(0)=-\mathrm{\infty }$ and ${\mathrm{\Phi }}^{-1}(1)=+\mathrm{\infty }$, and extend $\mathrm{\Phi }$ to $\bar{\mathbb{R}}$ by $\mathrm{\Phi }(-\mathrm{\infty })=0$ and $\mathrm{\Phi }(+\mathrm{\infty })=1$. With $Q_B$ from Lower empirical quantile of bootstrap values and $p=\alpha \mathrm{/}2$, return $[Q_B\text{\hspace{0.17em}⁣}(\mathrm{\Phi }(2z_0+{\mathrm{\Phi }}^{-1}(p))),Q_B\text{\hspace{0.17em}⁣}(\mathrm{\Phi }(2z_0+{\mathrm{\Phi }}^{-1}(1-p)))]\mathrm{.}$ On $\mathbb{R}$ the extended function is the usual standard-normal CDF. This is a finite-$B$ algorithm with an explicit tie rule; no coverage or higher-order accuracy claim is included.