Known mean, unknown spread, MLE for a Gaussian variance

Question

You model daily tracking errors of a hedged book as independent draws from $N(\mu, \sigma^2)$ , and by construction the mean is pinned at a known value $\mu$ (say the book is designed to be mean-zero). Only $\sigma^2$ is unknown.

Find the MLE of $\sigma^2$ and determine whether it is biased.

Accepted Answer

Likelihood. With \mu known, the log-likelihood in the parameter v = \sigma^2 is \ell(v) = -\frac{n}{2}\ln(2\pi v) - \frac{1}{2v}\sum{i=1}^n (xi - \mu)^2. Maximize. Differentiate with respect to v: \ell'(v) = -\frac{n}{2v} + \frac{1}{2v^2}\sumi (xi - \mu)^2 = 0 \implies \hat\sigma^2 = \frac{1}{n}\sum{i=1}^n (xi - \mu)^2. Bias. Here is the twist. Each (xi - \mu)^2 has expectation exactly \sigma^2 because \mu is the true known mean, so \mathbb{E}[\hat\sigma^2] = \frac{1}{n}\sumi \mathbb{E}[(xi-\mu)^2] = \sigma^2. Dividing by n is unbiased here. The usual n-1 correction only shows up when you mus…

Known mean, unknown spread, MLE for a Gaussian variance

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