*r*with Rayleigh distribution \(\sigma\) the MLE is \(\widehat{\sigma} = \frac{\sum r_i^2}{2n}\), and it is an unbiased estimator for \(\sigma\).

But any Monte Carlo test shows that's not true: Only the square root of that MLE is anything like an estimator for \(\sigma\).

Obviously I'm misunderstanding what it means to be a Maximum Likelihood Estimator. Can somebody explain what I'm missing?