The likelihood-ratio test compares the fit of a larger distribution family with a subset of the same family and determines whether there is a significant improvement in fit with the larger distribution. For instance, for a 2-parameter exponential distribution, the likelihood-ratio test compares the fit of 2-parameter exponential distribution family with the fit of 1-parameter exponential distribution family (a subset with the second parameter being 0). If a 2-parameter exponential distribution significantly improves the fit, then the p-value for likelihood-ratio test statistic is very small.
The likelihood-ratio test statistic is calculated as follows.
Let A be the maximum likelihood estimate (MLE) of the parameter vector for the larger distribution family (for example, the 3-parameter distribution family), and L(A) be the log likelihood. Let B be the MLE of the parameter vector for the corresponding smaller distribution family (for example, the corresponding 2-parameter distribution family), and L(B) be the log likelihood.
Likelihood-ratio test statistic = 2 * L(A) 2 * L(B).
Under the null hypothesis, the smaller distribution family fits the data well. The likelihood-ratio test statistic is chi-square distributed with df = dimension of vector (A) – dimension of vector (B).