Editing Likelihood-ratio test (section)

==Asymptotic distribution: Wilks’ theorem==
{{Main|Wilks' theorem}}

If the distribution of the likelihood ratio corresponding to a particular null and alternative hypothesis can be explicitly determined then it can directly be used to form decision regions (to sustain or reject the null hypothesis). In most cases, however, the exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine.{{Citation needed|date=September 2018}}

Assuming {{math|''H''<sub>0</sub>}} is true, there is a fundamental result by [[Samuel S. Wilks]]: As the sample size <math>n</math> approaches [[Infinity|<math>\infty</math>]], and if the null hypothesis lies strictly within the interior of the parameter space, the test statistic <math>\lambda_\text{LR}</math> defined above will be [[Asymptotic theory (statistics)|asymptotically]] [[chi-squared distribution|chi-squared distributed]] (<math>\chi^2</math>) with [[degrees of freedom (statistics)|degrees of freedom]] equal to the difference in dimensionality of <math>\Theta</math> and <math>\Theta_0</math>.<ref>{{cite journal |last=Wilks |first=S.S. |author-link=Samuel S. Wilks |doi=10.1214/aoms/1177732360 |title=The large-sample distribution of the likelihood ratio for testing composite hypotheses |journal=[[Annals of Mathematical Statistics]] |volume=9 |issue=1 |pages=60–62 |year=1938 |doi-access=free}}</ref> This implies that for a great variety of hypotheses, we can calculate the likelihood ratio <math>\lambda</math> for the data and then compare the observed <math>\lambda_\text{LR}</math> to the <math>\chi^2</math> value corresponding to a desired [[statistical significance]] as an ''approximate'' statistical test. Other extensions exist.{{which|date=March 2019}}