Difference between revisions of "Likelihood equation"
From Encyclopedia of Mathematics
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| − | An equation obtained by the [[ | + | 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 | + | 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|Sequential approximation, method of]]). |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) {{ZBL|0077.12901}}</TD></TR> | ||
| + | </table> | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.R. Cox, D.V. Hinkley, "Theoretical statistics" , Chapman | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> D.R. Cox, D.V. Hinkley, "Theoretical statistics" , Chapman and Hall (1974) pp. 21 {{ZBL|0334.62003}}</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 18:13, 19 November 2016
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).
References
| [1] | B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) Zbl 0077.12901 |
Comments
References
| [a1] | D.R. Cox, D.V. Hinkley, "Theoretical statistics" , Chapman and Hall (1974) pp. 21 Zbl 0334.62003 |
How to Cite This Entry:
Likelihood equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Likelihood_equation&oldid=19118
Likelihood equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Likelihood_equation&oldid=19118
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article