Efron (1981) attempted to improve the properties of the backwards method by adjusting the confidence coefficient to remove some of the bias from the method. The resulting method, called the bias-corrected method, uses an upper confidence limit equal to \(\hat{\theta}_{n}[\beta(\alpha)]=\) \(\hat{\theta}_{n}+n^{-1 / 2} \hat{\sigma}_{n} g_{\beta(\alpha)}\), where \(\beta(\alpha)=\Phi\left(z_{\alpha}+2 \tilde{\mu}ight)\) and \(\tilde{\mu}=\Phi^{-1}\left[G_{n}(0)ight]\).

a. Prove that \(\tilde{\mu}=n^{-1 / 2} r_{1}(0)+O\left(n^{-1}ight)\), as \(n ightarrow \infty\).

b. Prove that the bias-corrected critical point has the form \(\hat{\theta}_{n}[\beta(\alpha)]=\) \(\hat{\theta}_{n}+n^{-1 / 2} \hat{\sigma}_{n}\left\{z_{\alpha}+n^{-1 / 2}\left[2 r_{1}(0)-r_{1}\left(z_{\alpha}ight)ight]+O\left(n^{-1}ight)ight\}\), as \(n ightarrow \infty\).

c. Prove that the coverage probability of the bias-corrected upper confidence limit is given by \[\pi_{\mathrm{bc}}(\alpha)=\alpha+n^{-1 / 2}\left[2 r_{1}(0)-r_{1}\left(z_{\alpha}ight)-v_{1}\left(z_{\alpha}ight)ight] \phi\left(z_{\alpha}ight)+O\left(n^{-1}ight)\]
as \(n ightarrow \infty\).

d. Discuss the results given above in terms of the performance of this confidence interval.