Q2 (10 points)
Let $X_1, \dots, X_n$ be independent and identically distributed random variables. Assume $X_1$ has the Normal distribution mean with mean 0 and variance $\sigma^2 > 0$, that is,
$$ f(x; \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{x^2}{2\sigma^2}}, \quad \text{for } x \in \mathbb{R}, $$
where $\theta := \sigma^2 > 0$ is the parameter of interest.
(a). Compute the maximum likelihood estimator (MLE) $\hat{\theta}_n^{\text{MLE}}$.
(b). Compute the Fisher information at $\theta$ for $\theta \in (0, 1)$.
(c). Compute the Cramer-Rao lower bound for any unbiased estimators for $\theta \in (0, 1)$.
(d). Derive the limiting distribution of $\sqrt{n}(\hat{\theta}_n^{\text{MLE}} - \theta)$ as $n \to \infty$ for $\theta \in (0, 1)$.
(e). Construct a confidence interval based on the maximum likelihood estimator $\hat{\theta}_n^{\text{MLE}}$ that has an asymptotic coverage probability $1 - \alpha$ for some $\alpha \in (0, 1)$.
