Question: Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N(0, \theta)\) with unknown variance \(\theta>0\)

1. Find the maximum likelihood estimator \(\hat{\theta}\) of \(\theta\).
2. Find the method of moments estimator \(\tilde{\theta}\) of \(\theta\).
3. Are \(\hat{\theta}\) and \(\tilde{\theta}\) unbiased estimators of \(\theta ?\) Verify your answer.
4. Based \(\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2}\), construct a \((1-\alpha) 100 \%\) confidence interval for \(\theta\). Show and verify how to obtain a confidence interval for \(\theta\) with minimum length.

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