Wilks's theorem
In statistics, a likelihood ratio test (LR test) is a statistical test used for comparing the goodness of fit of two statistical models, one of which (the null model) is a special case of the other (the alternative model). The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other. This likelihood ratio, or equivalently its logarithm, can then be used to compute a p-value, or compared to a critical value to decide whether or not to reject the null model.
When the logarithm of the likelihood ratio is used, the statistic is known as a log-likelihood ratio statistic, and the probability distribution of this test statistic, assuming that the null model is true, can be approximated using Wilks’ theorem.
In the case of distinguishing between two models, each of which has no unknown parameters, use of the likelihood ratio test can be justified by the Neyman–Pearson lemma, which demonstrates that such a test has the highest power among all competitors.
A statistical model is often a parametrized family of probability density functions or probability mass functions . A simple-vs.-simple hypothesis test has completely specified models under both the null and alternative hypotheses, which for convenience are written in terms of fixed values of a notional parameter :
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