Normal behaviour
One interesting aspect of maximum-likelihood estimation is the common behaviour of estimators, regardless of the nature of the data and model. Recall that the maximum-likelihood estimate, \(\hat\theta\), is the value of a parameter \(\theta\) that maximises the likelihood function, \(L(\theta)\), or the log-likelihood function, \(\ell(\theta)=\log L(\theta)\). By way of example, consider the following three single-parameter distributions:
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