Definition:Maximum Likelihood Estimator

Definition

Let $\FF$ be a one-parameter family of probability distributions whose parameter is $\theta$.

Let $X$ be a continuous random variable belonging to a member of $\FF$.

Let $\map {\mathrm L} \theta$ be the likelihood function of $\theta$ with respect to $X$.

A maximum likelihood estimator is an estimator for $\theta$ to maximize $\map {\mathrm L} \theta$.

Properties

Let $\FF$ be a one-parameter family of probability distributions whose parameter is $\theta$.

Let $X$ be a continuous random variable belonging to a member of $\FF$.

Let $\EE$ be a maximum likelihood estimator for $\theta$ with respect to $X$.

Then $\EE$ is usually:

consistent

efficient

but not always unbiased.

Also see

• Results about maximum likelihood estimators can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): maximum likelihood estimation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): maximum likelihood estimation