2. You have a CAN estimator \hat{\theta} for parameter \theta, with point estimate \hat{\theta} = 0.45 and the asymptotic standard error aSE(\hat{\theta}) = 0.22. You are interested in the parameter \beta = exp(\theta).
(a) Find \hat{\beta}. Using the Delta method, find an asymptotic standard error aSE(\hat{\beta}) for \hat{\beta}.
(b) Using the above, calculate a 95\% asymptotic confidence interval for \beta.
(c) Using a t-test, find the P-value of the null hypothesis H_0: \beta = 1 against the alternative H_1: \beta \neq 1.
(d) Using a t-test, find the P-value of the null hypothesis H_0: \beta \leq 1 against the alternative H_1: \beta > 1.
(e) Calculate a 95\% asymptotic confidence interval [L, U] for the original parameter \theta. Calculate a 95\% asymptotic interval for \beta as [exp(L), exp(U)]. Compare this interval with your answer in (c).