The variance floor for a normal mean with known spread

Question

An instrument produces readings X1, \dots, Xn that are i.i.d. Normal with unknown mean \mu and known standard deviation \sigma. Compute the Fisher information for \mu, state the Cramer-Rao lower bound on the variance of an unbiased estimator of \mu, and check whether \bar X attains it.

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Score for one reading. The log-density is \log f(x;\mu) = -	frac12\log(2\pi\sigma^2) - \frac{(x-\mu)^2}{2\sigma^2}, so the score is s(x;\mu) = \frac{\partial}{\partial\mu}\log f = \frac{x-\mu}{\sigma^2}. Fisher information. It is the variance of the score: I(\mu) = \operatorname{Var}\!\left(\frac{X-\mu}{\sigma^2}ight) = \frac{\operatorname{Var}(X)}{\sigma^4} = \frac{\sigma^2}{\sigma^4} = \frac{1}{\sigma^2}. (The curvature form agrees: -\mathbb{E}[\partial^2\mu \log f] = 1/\sigma^2, a constant, since the second derivative is the non-random -1/\sigma^2.) The bound versus the sample mean. 	e…

The variance floor for a normal mean with known spread

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