Bias of the two variance estimators

Topic: Estimation · more questions on this topic
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Let X_1,\ldots,X_n be a random sample with variance \sigma^2, and write s^2 for the bias-corrected sample variance. The maximum likelihood estimator of \sigma^2 in the normal model is \widehat\sigma^2=((n-1)/n)s^2. Find the bias of \widehat\sigma^2, taking as known that \mathbb{E} s^2=\sigma^2.

Worked answer (click to show)

\mathbb{E}\widehat\sigma^2=\frac{n-1}{n}\mathbb{E} s^2=\frac{n-1}{n}\sigma^2.

The bias is \mathbb{E}\widehat\sigma^2-\sigma^2=-\sigma^2/n.