View source for Likelihood-ratio test

A statistical test based on the ratio of the greatest values of the likelihood functions under the hypothesis being tested and under all possible states of nature. Let a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588101.png" /> have values in the sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588103.png" />, let the family of measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588104.png" /> be absolutely continuous with respect to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588105.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588106.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588107.png" />. Suppose it is necessary, via a realization of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588108.png" />, to test the composite hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l0588109.png" /> according to which the unknown true value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881010.png" /> of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881011.png" /> belongs to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881012.png" />, against the composite alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881013.png" />. According to the likelihood-ratio test with significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881015.png" />, the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881016.png" /> has to be rejected if as a result of the experiment it turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881018.png" /> is the statistic of the likelihood-ratio test, defined by:

<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/l058810/l05881019.png" /></td> </tr></table>

while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881020.png" /> is the critical level determined by the condition that the size of the test,

<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/l058810/l05881021.png" /></td> </tr></table>

is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881022.png" />. In particular, if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881023.png" /> contains only two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881024.png" />, with densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881026.png" /> respectively, corresponding to the concurrent hypotheses which, in this case, are simple, then the statistic of the likelihood-ratio test is simply

<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/l058810/l05881027.png" /></td> </tr></table>

According to the likelihood-ratio test with significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881028.png" />, the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881029.png" /> has to be rejected if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881030.png" />, where the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881032.png" />, is determined by the condition

<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/l058810/l05881033.png" /></td> </tr></table>

<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/l058810/l05881034.png" /></td> </tr></table>

The (generalized) likelihood-ratio test was proposed by J. Neyman and E.S. Pearson in 1928. They also proved (1933) that of all level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058810/l05881035.png" /> tests for testing one simple hypothesis against another, the likelihood-ratio test is the most powerful (see [[Neyman–Pearson lemma|Neyman–Pearson lemma]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Neyman,   E.S. Pearson,   "Joint statistical papers" , Cambridge Univ. Press  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>



====Comments====
This test is also called the generalized likelihood-ratio test, or the Wald test.