Order selection in finite mixture models: complete or observed likelihood information criteria?

Choosing the number of components in a finite mixture model is a challenging task. In this article, we study the behaviour of information criteria for selecting the mixture order, based on either the observed likelihood or the complete likelihood including component labels. We propose a new observed likelihood criterion called AiCmix, which is shown to be order consistent. We further show that when there is a nontrivial level of classification uncertainty in the true model, complete likelihood criteria asymptotically underestimate the true number of components. A simulation study illustrates the potentially poor finitesample performance of complete likelihood criteria, while AICmix and the Bayesian information criterion perform strongly regardless of the level of classification uncertainty.