Editing Likelihood-ratio test (section)

===Case of simple hypotheses===
{{Main|Neyman–Pearson lemma}}
A simple-vs.-simple hypothesis test has completely specified models under both the null hypothesis and the alternative hypothesis, which for convenience are written in terms of fixed values of a notional parameter <math>\theta</math>:

:<math>
\begin{align}
H_0 &:& \theta=\theta_0 ,\\
H_1 &:& \theta=\theta_1 .
\end{align}
</math>
In this case, under either hypothesis, the distribution of the data is fully specified: there are no unknown parameters to estimate. For this case, a variant of the likelihood-ratio test is available:<ref>{{cite book |last1=Mood |first1=A.M. |last2=Graybill |first2=F.A. |first3=D.C. |last3=Boes |year=1974 |title=Introduction to the Theory of Statistics |edition=3rd |publisher=[[McGraw-Hill]] |at=§9.2}}</ref><ref name="Stuart et al. 20.10–20.13">{{citation |last1=Stuart|first1=A. |last2=Ord |first2=K. |last3=Arnold |first3=S. |year=1999 |title=Kendall's Advanced Theory of Statistics |volume=2A |publisher=[[Edward Arnold (publisher)|Arnold]] |at=§§20.10–20.13}}</ref>

:<math>
\Lambda(x) = \frac{~\mathcal{L}(\theta_0\mid x) ~}{~\mathcal{L}(\theta_1\mid x) ~}.
</math>

Some older references may use the reciprocal of the function above as the definition.<ref>{{citation |author1-last=Cox |author1-first=D. R.  |author1-link= David Cox (statistician)|author2-last=Hinkley |author2-first=D. V. | author2-link= David Hinkley |title=Theoretical Statistics |publisher= [[Chapman & Hall]] |year=1974 |isbn=0-412-12420-3 |page=92 }}</ref> Thus, the likelihood ratio is small if the alternative model is better than the null model.

The likelihood-ratio test provides the decision rule as follows:

:If <math>~\Lambda > c ~</math>, do not reject <math>H_0</math>;
:If <math>~\Lambda < c ~</math>, reject <math>H_0</math>;
:If <math>~\Lambda = c ~</math>, reject <math>H_0</math> with probability <math>~q~</math>.
:
The values <math>c</math> and <math>q</math> are usually chosen to obtain a specified [[significance level]] <math>\alpha</math>, via the relation
:<math>~q~</math> <math> \operatorname{P}(\Lambda=c \mid H_0)~+~\operatorname{P}(\Lambda < c \mid H_0)~=~\alpha~. </math>
The [[Neyman–Pearson lemma]] states that this likelihood-ratio test is the [[Statistical power|most powerful]] among all level <math>\alpha</math> tests for this case.<ref name="NeymanPearson1933"/><ref name="Stuart et al. 20.10–20.13"/>