Consider N(mu, sigma^2) distribution. Determine the Fisher information I(sigma^2

Question

Consider N(mu, sigma^2) distribution. Determine the Fisher information I(sigma^2). That is, consider N (mu, theta) distribution. Determine the Fisher information I(theta). Let X_1, X_2,..., X_n be a random sample of size n from a N([mu, sigma^2) distribution. Suppose that mu is unknown. We know that the sample variance S^2 is an unbiased estimator of sigma^2. Is S^2 an efficient estimator of sigma^2? If not, find its efficiency. Recall that (n - 1) S^2/sigma^2 has chi^2(n - 1) distribution. Recall that E(chi_2(r)) = 2r, Suppose that mu is known. Find the maximum likelihood estimator for sigma^2. Is the maximum likelihood estimator for sigma^2 unbiased? Justify your answer. Is the maximum likelihood estimator for sigma^2 efficient? If not, find its efficiency.

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Consider N(mu, sigma^2) distribution. Determine the Fisher information I(sigma^2

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