Model Selection for Mixture Models Using Perfect Sample

We have considered a perfect sample method for model selection of finite mixture models with either known (fixed) or unknown number of components which can be applied in the most general setting with assumptions on the relation between the rival models and the true distribution. It is, both, one or neither to be well-specified or mis-specified, they may be nested or non-nested. We consider mixture distribution as a complete-data (bivariate) distribution by prediction of missing data variable (unobserved variable) and show that this ideas is applicable to use Vuong's test for select optimum mixture model when number of components are known (fixed) or unknown. We have considered  AIC  and  BIC  based on the complete-data distribution. The performance of this method is evaluated by Monte-Carlo method and real data set, as Total Energy Production.