Likelihood-ratio Test - Definition (likelihood Ratio Test For Composite Hypotheses)

Definition (likelihood Ratio Test For Composite Hypotheses)

A null hypothesis is often stated by saying the parameter is in a specified subset of the parameter space .

\begin{align} H_0 &:& \theta \in \Theta_0\\ H_1 &:& \theta \in \Theta_0^{\complement} \end{align}

The likelihood function is (with being the pdf or pmf) is a function of the parameter with held fixed at the value that was actually observed, i.e., the data. The likelihood ratio test statistic is

Here, the notation refers to the Supremum function.

A likelihood ratio test is any test with critical region (or rejection region) of the form where is any number satisfying . Many common test statistics such as the Z-test, the F-test, Pearson's chi-squared test and the G-test are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof.

Read more about this topic:  Likelihood-ratio Test

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