Editing Maximum likelihood estimation (section)

=== Efficiency ===
As assumed above, if the data were generated by <math>~f(\cdot\,;\theta_0)~,</math> then under certain conditions, it can also be shown that the maximum likelihood estimator [[Convergence in distribution|converges in distribution]] to a normal distribution. It is {{sqrt|''n''&nbsp;}}-consistent and asymptotically efficient, meaning that it reaches the [[Cramér–Rao bound]]. Specifically,<ref name=":1"/>

<math display="block">
    \sqrt{n\,} \, \left( \widehat{\theta\,}_\text{mle} - \theta_0 \right)\ \ \xrightarrow{d}\ \ \mathcal{N} \left( 0,\ \mathcal{I}^{-1} \right) ~,
  </math>
where <math>~\mathcal{I}~</math> is the [[Fisher information matrix]]:
<math display="block">
    \mathcal{I}_{jk} = \operatorname{\mathbb E} \, \biggl[ \; -{ \frac{\partial^2\ln f_{\theta_0}(X_t)}{\partial\theta_j\,\partial\theta_k } } 
             \; \biggr] ~.
  </math>

In particular, it means that the [[bias of an estimator|bias]] of the maximum likelihood estimator is equal to zero up to the order {{sfrac|1|{{sqrt|{{mvar|n}}&nbsp;}}}}.