Cramer-Rao bound for a normal variance

Question

Let X1, \dots, Xn be i.i.d. Normal with known mean \mu and unknown variance v = \sigma^2. Compute the Fisher information for v, state the Cramer-Rao bound, and find an efficient unbiased estimator.

Accepted Answer

Score in terms of v = \sigma^2. The normal log-density of one observation is \log f(x; v) = -\frac12\log(2\pi v) - \frac{(x - \mu)^2}{2v}. Differentiating with respect to v: s(x; v) = -\frac{1}{2v} + \frac{(x - \mu)^2}{2v^2} = \frac{(x - \mu)^2 - v}{2v^2}. (Sanity check: E[(X - \mu)^2] = v, so E[s] = 0.) Fisher information. For a normal, (X - \mu)^2 = v\,Z^2 with Z standard normal, and \operatorname{Var}(Z^2) = 2, so \operatorname{Var}\big((X-\mu)^2\big) = 2v^2. Then I(v) = \operatorname{Var}\!\left(\frac{(X - \mu)^2 - v}{2v^2}\right) = \frac{2v^2}{4v^4} = \frac{1}{2v^2} = \frac{1}{2\sigma^4}…

Cramer-Rao bound for a normal variance

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