(f) (10%) How does (a) to (e) give you any thought on the estimation for \(\theta\)? Briefly describe your thoughts, and justify your thoughts.

2. (50%) Let $X1, X2, \cdots, X_n$ be a random sample from a normal distribution $$f(x; \theta) = (2\pi\theta)^{-1/2} \exp \left( -\frac{1}{2\theta} (x - \mu)^2 \right), x \in R; \mu \in R, \theta > 0.$$ Let the distribution function of $f(x; \theta)$ be denoted as $\Phi(x; \mu, \theta)$. (a) (10%) Find the maximum likelihood estimators \(\hat{\mu},\hat{\theta}\) of \(\mu ,\theta\)

(b) (10%) Find the Rao-Cramer lower bound of \(\hat{\mu}\) and \(\hat{\theta}\), respectively.

(c) (10%) Find the asymptotic distribution of \(\hat{\mu},\hat{\theta}\)

(d) (10%) Find the asymptotic distribution of \(\Phi(x; \hat{\mu}, \hat{\theta})\). [Hint: Use Delta method and the derivatives with respect to \(\mu\) and \(\theta\) do not need to be expanded.]

(e) (10%) Find the Likelihood ratio test (LR test) for testing \( H0: \theta = \theta0 \) versus \( H0: \theta \neq \theta0 \), where \( \theta_0 \) is a pre-specified value and provide the corresponding critical value for the LR test.

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