Consider a sequence of independent and identically distributed \(d\)-dimensional random vectors \(\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}\) from a \(d\)-dimensional distribution \(F\). Assume the structure smooth function model with \(\boldsymbol{\mu}=E\left(\mathbf{X}_{n}ight), \theta=g(\boldsymbol{\mu})\) with \(\hat{\theta}_{n}=g\left(\overline{\mathbf{X}}_{n}ight)\). Further, assume that

\[\sigma^{2}=h^{2}(\mu)=\lim _{n ightarrow \infty} V\left(n^{1 / 2} \hat{\theta}ight)\]

with \(\hat{\sigma}_{n}^{2}=h^{2}\left(\overline{\mathbf{X}}_{n}ight)\). Let \(G_{n}(t)=P\left[n^{1 / 2} \sigma^{-1}\left(\hat{\theta}_{n}-\thetaight) \leq tight]\) and \(H_{n}(t)=\) \(P\left[n^{1 / 2} \hat{\sigma}_{n}^{-1}\left(\hat{\theta}_{n}-\thetaight) \leq tight]\) and define \(g_{\alpha, n}\) and \(h_{\alpha, n}\) to be the corresponding \(\alpha\) quantiles of \(G_{n}\) and \(H_{n}\). Define the ordinary and studentized \(100 \alpha \%\) upper confidence limits for \(\theta\) as \(\hat{\theta}_{n, \text { ord }}=\hat{\theta}_{n}-n^{-1 / 2} \sigma g_{1-\alpha}\) and \(\hat{\theta}_{n, \text { stud }}=\hat{\theta}_{n}-\) \(n^{-1 / 2} \hat{\sigma}_{n} h_{1-\alpha}\). Prove that \(\hat{\theta}_{n, \text { ord }}\) and \(\hat{\theta}_{n, \text { stud }}\) are accurate upper confidence limits.