You fit a model with 3 parameters and obtain a log-likelihood of -1042.0. Adding 2 more parameters gives -1038.5. Test the restriction at the 5% level with the likelihood ratio statistic, give the p-value, and state the conditions the asymptotic distribution requires.

You fit a model with 3 parameters and obtain a log-likelihood of -1042.0. Adding 2 more parameters gives -1038.5. Test the restriction at the 5% level with the likelihood ratio statistic, give the p-value, and state the conditions the asymptotic distribution requires.

Approach: Form twice the log-likelihood difference and compare it with the chi-square distribution whose degrees of freedom equal the number of restrictions.

7.0. The likelihood ratio statistic is 2*(l_full - l_restricted) = 2*(-1038.5 + 1042.0) = 7.0. Under the null that the two extra parameters are zero, Wilks' theorem gives an asymptotic chi-square distribution with 2 degrees of freedom, whose 5% critical value is 5.99, so the restriction is rejected. With 2 degrees of freedom the survival function has the closed form exp(-x/2), giving a p-value of exp(-3.5) = 0.030. The asymptotic result needs the models to be nested, the null parameter value to lie in the interior of the parameter space, the information matrix to be non-singular there, and the usual smoothness of the likelihood. Testing a variance component at zero or the number of components in a mixture violates the interior condition, and the chi-square reference is then wrong in a direction that over-rejects.

Follow-up: The two models are non-nested and fitted on the same data. What test replaces the likelihood ratio, and what does it compare?

Key concepts: likelihood ratio test, Wilks theorem, degrees of freedom, nested models.