The large-sample spread of an estimated variance

Question

Readings X1, \dots, Xn are i.i.d. Normal with known mean \mu and unknown variance 	heta = \sigma^2. The maximum likelihood estimator is \hat	heta = \frac{1}{n}\sumi (Xi - \mu)^2. Find the large-sample distribution of \hat	heta and confirm it is asymptotically efficient. What is its asymptotic variance factor?

Accepted Answer

General result. The maximum likelihood estimator obeys \sqrt{n}\,(\hat	heta - 	heta) \xrightarrow{d} N\!\left(0, \frac{1}{I(	heta)}ight). Fisher information for the variance. For a normal with known mean, the information about 	heta = \sigma^2 per observation is I(	heta) = \frac{1}{2\sigma^4}, \qquad \frac{1}{I(	heta)} = 2\sigma^4. Hence \sqrt{n}\,(\hat	heta - 	heta) \xrightarrow{d} N(0,\ 2\sigma^4), so the asymptotic variance factor is 2\sigma^4. Direct check. \hat	heta is an average of the i.i.d. terms (Xi - \mu)^2. Each term, divided by \sigma^2, is a chi-square with one degree …

The large-sample spread of an estimated variance

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