View source for Likelihood equation

An equation obtained by the [[Maximum-likelihood method|maximum-likelihood method]] when finding statistical estimators of unknown parameters. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058800/l0588001.png" /> be a random vector the probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058800/l0588002.png" /> of which contains an unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058800/l0588003.png" />. Then the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058800/l0588004.png" /></td> </tr></table>

is called the likelihood equation and its solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058800/l0588005.png" /> is called the maximum-likelihood estimator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058800/l0588006.png" />. 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,   "Mathematische Statistik" , Springer  (1957)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.R. Cox,   D.V. Hinkley,   "Theoretical statistics" , Chapman &amp; Hall  (1974)  pp. 21</TD></TR></table>