In the context of the development of the bias corrected and accelerated bootstrap confidence interval, prove that

\[\hat{m}+\left(\hat{m}+z_{\alpha}ight)\left[1-\hat{\alpha}\left(\hat{m}+z_{\alpha}ight)ight]^{-1}=z_{\alpha}+2 \hat{m}+z_{\alpha}^{2} \hat{a}+O_{p}\left(n^{-1}ight),\]

as \(n ightarrow \infty\). This form of the expression is of the same form given by Efron (1987).