Maximum likelihood estimator

Properties

• Consistency: the sequence of MLEs converges in probability to the value being estimated.
• Equivariance: If ${\displaystyle {\hat {\theta }}}$ is the maximum likelihood estimator for ${\displaystyle \theta }$, and if ${\displaystyle g(\theta )}$ is a bijective transform of ${\displaystyle \theta }$, then the maximum likelihood estimator for ${\displaystyle \alpha =g(\theta )}$ is ${\displaystyle {\hat {\alpha }}=g({\hat {\theta }})}$. The equivariance property can be generalized to non-bijective transforms, although it applies in that case on the maximum of an induced likelihood function which is not the true likelihood in general.
• Efficiency, i.e. it achieves the Cramér–Rao lower bound when the sample size tends to infinity. This means that no consistent estimator has lower asymptotic mean squared error than the MLE (or other estimators attaining this bound), which also means that MLE has asymptotic normality.
• Second-order efficiency after correction for bias.