For n iid N(mu, sigma^2) observations with mu unknown, compute the Cramer-Rao bound for unbiased estimators of sigma^2, give the exact variance of the sample variance s^2, and say whether the bound is attained and what follows from the answer.

For n iid N(mu, sigma^2) observations with mu unknown, compute the Cramer-Rao bound for unbiased estimators of sigma^2, give the exact variance of the sample variance s^2, and say whether the bound is attained and what follows from the answer.

Approach: Parameterise by the variance itself, differentiate the log density twice in that parameter, and compare the resulting bound with the chi-square variance of the sample variance.

2*sigma^4/n. Write v = sigma^2. The log density of one observation is -log(2*pi*v)/2 - (x - mu)^2/(2*v), so the score in v is -1/(2*v) + (x - mu)^2/(2*v^2), and using Var((x - mu)^2) = 2*v^2 its variance is the Fisher information 1/(2*v^2). The information matrix in (mu, v) is diagonal, so estimating mu costs nothing asymptotically and the bound for unbiased estimators of v is 1/(n/(2*v^2)) = 2*sigma^4/n. The sample variance satisfies that (n - 1)*s^2 is sigma^2 times a chi-square with n - 1 degrees of freedom, so Var(s^2) = 2*sigma^4/(n - 1), strictly above the bound at every finite n. Attainment would require the score to be an affine function of the estimator, which happens only when mu is known and the estimator uses it. The ratio n/(n - 1) tends to 1, so s^2 is asymptotically efficient even though it is inefficient at any fixed n.

Follow-up: The observations have excess kurtosis 3 instead of 0. What happens to Var(s^2) and to the Cramer-Rao bound computed under the normal model?

Key concepts: Cramer-Rao bound, Fisher information, score function, asymptotic efficiency.