Consider the framework of the smooth function model where \(\sigma\), which denotes the asymptotic variance of \(n^{1 / 2} \hat{\theta}_{n}\), is unknown, and the test statistic \(T_{n}=n^{1 / 2} \hat{\sigma}_{n}^{-1}\left(\hat{\theta}_{n}-\theta_{0}ight)\) follows the distribution \(H_{n}\) when \(\theta_{0}\) is the true value of \(\theta\).

a. Prove that a test of the null hypothesis \(H_{0}: \theta \leq \theta_{0}\) against the alternative hypothesis \(H_{1}: \theta>\theta_{0}\) that rejects the null hypothesis if \(T_{n}>z_{1-\alpha}\) is a first-order accurate test.

b. Prove that a test of the null hypothesis \(H_{0}: \theta \leq \theta_{0}\) against the alternative hypothesis \(H_{1}: \theta>\theta_{0}\) that rejects the null hypothesis if \(T_{n}>z_{1-\alpha}+n^{-1 / 2} \hat{s}_{1}\left(z_{1-\alpha}ight)\) is a second-order accurate test.