Consider the framework of the smooth function model where \(\sigma\), which denotes the asymptotic variance of \(n^{1 / 2} \hat{\theta}_{n}\), is known. Consider using the test statistic \(Z_{n}=n^{1 / 2} \sigma^{-1}\left(\hat{\theta}_{n}-\theta_{0}ight)\) which follows the distribution \(G_{n}\) when \(\theta_{0}\) is the true value of \(\theta\). Prove that an unbiased test of size \(\alpha\) of the null hypothesis \(H_{0}: \theta \leq \theta_{0}\) against the alternative hypothesis \(H_{1}: \theta>\theta_{0}\) rejects the null hypothesis if \(Z_{n}>g_{1-\alpha}\), where we recall that \(g_{1-\alpha}\) is the \((1-\alpha)^{\text {th }}\) quantile of the distribution \(G_{n}\).