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 $ X $ have values in the sample space $ \{ \mathfrak X , {\mathcal B} , {\mathsf P} _ \theta \} $, $ \theta \in \Theta $, let the family of measures $ {\mathcal P} = \{ { {\mathsf P} _ \theta } : {\theta \in \Theta } \} $ be absolutely continuous with respect to a $ \sigma $- finite measure $ \mu $ and let $ p _ \theta ( x) = d {\mathsf P} _ \theta ( x)/d \mu ( x) $. Suppose it is necessary, via a realization of the random variable $ X $, to test the composite hypothesis $ H _ {0} $ according to which the unknown true value $ \theta _ {0} $ of the parameter $ \theta $ belongs to the set $ \Theta _ {0} \subset \Theta $, against the composite alternative $ H _ {1} : \theta _ {0} \in \Theta _ {1} = \Theta \setminus \Theta _ {0} $. According to the likelihood-ratio test with significance level $ \alpha $, $ 0 < \alpha < 1/2 $, the hypothesis $ H _ {0} $ has to be rejected if as a result of the experiment it turns out that $ \lambda ( x) \leq \lambda _ \alpha $, where $ \lambda ( X) $ is the statistic of the likelihood-ratio test, defined by:

$$ \lambda ( X) = \frac{\sup _ {\theta \in \Theta _ {0} } \ p _ \theta ( X) }{\sup _ {\theta \in \Theta } p _ \theta ( X) } , $$

while $ \lambda _ \alpha $ is the critical level determined by the condition that the size of the test,

$$ \sup _ {\theta \in \Theta _ {0} } {\mathsf P} _ \theta \{ \lambda ( x) \leq \lambda _ \alpha \} = \ \sup _ {\theta \in \Theta _ {0} } \ \int\limits _ {\{ {x } : {\lambda ( x) \leq \lambda _ \alpha } \} } p _ \theta ( x) \mu ( dx) , $$

is equal to $ \alpha $. In particular, if the set $ \Theta $ contains only two points $ \Theta = \{ {\mathsf P} _ {0} , {\mathsf P} _ {1} \} $, with densities $ p _ {0} ( \cdot ) $ and $ p _ {1} ( \cdot ) $ respectively, corresponding to the concurrent hypotheses which, in this case, are simple, then the statistic of the likelihood-ratio test is simply

$$ \lambda ( X) = \ \frac{p _ {0} ( X) }{\max \{ p _ {0} ( X), p _ {1} ( X) \} } = \ \min \left \{ 1, \frac{p _ {0} ( X) }{p _ {1} ( X) } \right \} . $$

According to the likelihood-ratio test with significance level $ \alpha $, the hypothesis $ H _ {0} $ has to be rejected if $ p _ {0} ( X)/p _ {1} ( X) \leq \lambda _ \alpha $, where the number $ \lambda _ \alpha $, $ 0 < \lambda _ \alpha < 1 $, is determined by the condition

$$ {\mathsf P} \{ \lambda ( X) < \lambda _ \alpha \mid H _ {0} \} = $$

$$ = \ \int\limits _ {\{ x: p _ {0} ( x) \leq p _ {1} ( x) \lambda _ \alpha \} } p _ {0} ( x) \mu ( dx) = \alpha . $$

The (generalized) likelihood-ratio test was proposed by J. Neyman and E.S. Pearson in 1928. They also proved (1933) that of all level- $ \alpha $ tests for testing one simple hypothesis against another, the likelihood-ratio test is the most powerful (see Neyman–Pearson lemma).

References

[1]	J. Neyman, E.S. Pearson, "Joint statistical papers" , Cambridge Univ. Press (1967)
[2]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)

Comments

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