Maximum Likelihood Estimator from Likelihood Function

Theorem

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 $X$.

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

Then $\EE$ can be found by:

calculating $\theta$ for which $\map {\dfrac \d {\d \theta} } {\map \ln {\map {\mathrm L} \theta} } = 0$

determine which of these solutions corresponds to a maximum.

It is often the case that an iterative solution is needed.

Proof

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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