Bias-corrected and accelerated bootstrap interval

Assume AC. Input observed estimate $\widehat{\theta }$, a nonempty list of $B$ bootstrap estimates, $n\ge 2$ delete-one estimates, and $0<\alpha <1$. The bootstrap estimates are generated under IID empirical bootstrap sample and mean statistic. Compute ${\widehat{p}}_0=\frac{\mathrm{\#}\{b:{\widehat{\theta }}_b^{\ast }<\widehat{\theta }\}+\frac{1}{2}\mathrm{\#}\{b:{\widehat{\theta }}_b^{\ast }=\widehat{\theta }\}}{B},{\widehat{z}}_0={\mathrm{\Phi }}^{-1}({\widehat{p}}_0),$ using the midrank tie rule and extended endpoint convention ${\mathrm{\Phi }}^{-1}(0)=-\mathrm{\infty }$, ${\mathrm{\Phi }}^{-1}(1)=+\mathrm{\infty }$, and compute ${\widehat{a}}_J$ by Delete-one jackknife acceleration.

If ${\widehat{p}}_0\in \{0,1\}$ or ${\widehat{a}}_J$ is undefined, return undefined. Otherwise, for $\gamma \in \{\alpha \mathrm{/}2,1-\alpha \mathrm{/}2\}$ set $z_{\gamma }={\mathrm{\Phi }}^{-1}(\gamma ),p_{\gamma }=\mathrm{\Phi }\text{\hspace{0.17em}⁣}(z_0+\frac{z_0+z_{\gamma }}{1-{\widehat{a}}_J(z_0+z_{\gamma })}),$ where $z_0={\mathrm{\Phi }}^{-1}({\widehat{p}}_0)$. If either denominator is zero or $p_{\alpha \mathrm{/}2}\ge p_{1-\alpha \mathrm{/}2}$, return undefined. Otherwise return $[Q_B(p_{\alpha \mathrm{/}2}),Q_B(p_{1-\alpha \mathrm{/}2})]$, using Lower empirical quantile of bootstrap values. These instructions define a finite-sample procedure only; no BCa coverage theorem is asserted.