The variance floor for a normal variance with known mean

Question

Readings X1, \dots, Xn are i.i.d. Normal with a known mean \mu but unknown variance 	heta = \sigma^2. Treat 	heta itself as the parameter. Compute the Fisher information for 	heta, state the Cramer-Rao lower bound for an unbiased estimator of 	heta, and check whether the natural estimator attains it.

Accepted Answer

Score in terms of 	heta. With mean known, the log-density of one reading is \log f(x;	heta) = -	frac12\log(2\pi	heta) - \frac{(x-\mu)^2}{2	heta}, so the score is s(x;	heta) = -\frac{1}{2	heta} + \frac{(x-\mu)^2}{2	heta^2}. Fisher information by curvature. Differentiate again: \frac{\partial^2}{\partial	heta^2}\log f = \frac{1}{2	heta^2} - \frac{(x-\mu)^2}{	heta^3}. Take minus the expectation, using \mathbb{E}[(X-\mu)^2] = 	heta: I(	heta) = -\frac{1}{2	heta^2} + \frac{	heta}{	heta^3} = -\frac{1}{2	heta^2} + \frac{1}{	heta^2} = \frac{1}{2	heta^2} = \frac{1}{2\sigma^4}. The bo…

The variance floor for a normal variance with known mean

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