Root selection in normal mixture models

Finite mixtures of normal distributions are attractive in identifying the underlying group structure in the data. However, it is a challenging task to do statistical inference in normal mixture models using the method of maximum likelihood, due to the unbounded likelihood and the existence of multiple roots to the likelihood equation including a so-called spurious root. In this article we propose a new likelihood-based method for selecting a statistically reasonable root when there exist multiple roots of the likelihood equation for a finite normal mixture model. We first prove that our proposed methodology can choose a root to the mixture likelihood equation with consistency. We then show, by simulation studies and real examples, that the proposed methods can greatly reduce the risk of choosing problematic roots that have the same features as spurious roots.