MLE of a normal variance when the mean is known

Question

You observe X1, \dots, Xn i.i.d. from \mathcal{N}(\mu0, \sigma^2) where the mean \mu0 is known and only \sigma^2 is unknown. Derive the MLE of \sigma^2. Is it biased? Contrast with the unknown-mean case.

Accepted Answer

Log-likelihood (only \sigma^2 is free; write v = \sigma^2): \ell(v) = -\frac{n}{2}\log(2\pi) - \frac{n}{2}\log v - \frac{1}{2v}\sum{i=1}^n (Xi - \mu0)^2. Maximize over v. Differentiate and set to zero: \frac{\partial \ell}{\partial v} = -\frac{n}{2v} + \frac{1}{2v^2}\sum{i=1}^n (Xi - \mu0)^2 = 0 \implies \hat\sigma^2 = \frac{1}{n}\sum{i=1}^n (Xi - \mu0)^2. Note the spread is measured around the true mean \mu0, not an estimated one. Bias check. Each term has \mathbb{E}[(Xi - \mu0)^2] = 	ext{Var}(Xi) = \sigma^2, so \mathbb{E}[\hat\sigma^2] = \frac{1}{n}\sum{i=1}^n \mathbb{E}[(Xi - \mu0)^2] = \…

MLE of a normal variance when the mean is known

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