Maximum Likelihood Estimator (mle)

Definition

Let

• $\mathbf{X}$ : Observed data
• $\theta$ : Parameter
• $L(\theta ;\mathbf{X})$ : Likelihood function

If $\hat{\theta }=\mathrm{Argmax}L(\theta ;\mathbf{X})$

Then $\hat{\theta }$ is a maximum likelihood estimator (mle) of $\theta$

Properties

The MLE has the following properties:

1. Consistenty $\hat{\theta }\stackrel{P}{\to }\theta \text{ketika }n\to \infty$
2. Asymptotic normality $\sqrt{n}(\hat{\theta }-\theta )\stackrel{d}{\to }N(0,V)$ Where $V$ is covariance matrix.
3. Asymptotic efficiency $V=I(\theta {)}^{-1}$ Where $I(\theta {)}^{-1}$ is Rao-Cramer Lower Bound

See also

• Finding Maximum Likelihood Estimator

Related theorems

• Theorem 6.1.2