Prove that the coverage probability of a \(100 \alpha \%\) upper confidence limit that has an asymptotic expansion of the form

\[\hat{\theta}_{n}(\alpha)=\hat{\theta}_{n}+n^{-1 / 2} \hat{\sigma}_{n} z_{\alpha}+n^{-1} \hat{\sigma}_{n} \hat{s}_{1}\left(z_{\alpha}ight)+n^{-3 / 2} \hat{\sigma}_{n} \hat{s}_{2}\left(z_{\alpha}ight)+O_{p}\left(n^{-2}ight),\]

has an asymptotic expansion of the form

\[\pi_{n}(\alpha)=n^{-1} u_{\alpha} z_{\alpha} \phi\left(z_{\alpha}ight)+O\left(n^{-3 / 2}ight),\]

as \(n ightarrow \infty\). Is there any case where \(u_{\alpha}=0\) so that the resulting confidence limit would be third-order accurate?