Likelihood equation

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An equation obtained by the maximum-likelihood method when finding statistical estimators of unknown parameters. Let $X$ be a random vector for which the probability density $p(x|\theta)$ contains an unknown parameter $\theta \in \Theta$. Then the equation $$ \frac{\partial}{\partial\theta} \log p(X|\theta) = 0 $$ is called the likelihood equation and its solution $\hat\theta$ is called the maximum-likelihood estimator for $\theta$. In some cases the likelihood equation can be solved in an elementary way. However, in general, the likelihood equation is an algebraic or transcendental equation, solved by the method of successive approximation (cf. Sequential approximation, method of).


[1] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) Zbl 0077.12901



[a1] D.R. Cox, D.V. Hinkley, "Theoretical statistics" , Chapman and Hall (1974) pp. 21 Zbl 0334.62003
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Likelihood equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article