Vuong's closeness test

Vuong the test is a statistical test for model selection, which is based on the Bayesian Information Criterion. It is named after the mathematician Quang H. Vuong, who suggested the test in 1989.

Theoretical procedure

The Vuong test tests the null hypothesis that two models - whether they are hierarchical, non-hierarchical or overlapping - the same lie close to the true distribution against the alternative hypothesis that a model is closer to it. But he does not state that the better model is really the true model. Assuming non-hierarchical and identical and independently distributed explanatory variables model is 1 (or Model 2 ) are preferred on the significance level if the test statistic

With

The (negative ) - quantile of the standard normal distribution exceeds ( or falls below ). The counter variable is the similar to the Bayesian information criterion for the number of coefficients corrected difference between the maximum log - likelihoods of the model estimates the denominator size corresponds to the sum of the squares of

For hierarchical and overlapping models, the test statistic is

Compared with the corresponding critical sizes from a weighted sum of chi-square distributions. This can be approximated by a gamma distribution:

With

Here, the vector is the eigenvalues ​​of a matrix related expected values ​​. Its derivation is quite difficult, so that statements are made in the overlapping case, mostly due to subjectively sufficiently large values ​​.