Model Selection in the Case of Mixture of Normal Densities

In this paper we illustrate a procedure in finding the number of the components of a mixture of bivariate normal densities, exploiting the properties of robustness of the estimates based on the Minimum L_2 distance. Each step of the procedure consists in the comparison between the estimates, according to Maximum Likelihood and Minimum L_2 criteria, of a mixture with a fixed number of components. The discrepancy between the two estimated densities is measured applying the concept of similarity between densities. A test of statistical hypothesis, based on Monte Carlo Significance Test, is introduced to verify the similarity between the two estimates. If their dissimilarity may be judged significant, then we change the model adding one more component to the mixture.