# 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)