The Vuong (1989) and Clarke (2007) tests are likelihood-ratio-based tests for model selection that use the Kullback-Leibler information criterion.
The implemented tests can be used for choosing between two bivariate models which are non necessary nested.
In the Vuong test, the null hypothesis is that the two models are equally close to the actual model, whereas the alternative is that one model is closer. The test follows asymptotically a standard normal distribution under the null. Assume that the critical region is (-c,c), where c is typically set to 1.96. If the value of the test is greater than c then we reject the null hypothesis that the models are equivalent in favour of the model in obj1. Vice-versa if the value is smaller than -c we reject the null hypothesis that the models are equivalent in favour of the model in obj2. If the value falls within (-c,c0) then we cannot discriminate between the two competing models given the data.
In the Clarke test, if the two models are statistically equivalent then the log-likelihood ratios of the observations should be evenly distributed around zero and around half of the ratios should be larger than zero. The test follows asymptotically a binomial distribution with parameters n and 0.5. Critical values can be obtained as shown in Clarke (2007). Intuitively, the model in obj1 is preferred over that in obj2 if the value of the test is significantly larger than its expected value under the null hypothesis ('coden/2), and vice versa. If the value is not significantly different from n/2 then obj1 can be thought of as equivalent to obj2.