A classical invariance approach to the normal mixture problem

Although normal mixture models have received great attention and are commonly used in different fields, they stand out for failing to have a finite maximum on the likelihood. In the univariate case, there are n solutions, corresponding to n distinct data points, along a parameter boundary, each with an infinite spike of the likelihood, and none making particular sense as a chosen solution. The multivariate case yields an even more complex likelihood surface. In this paper, we show that there is a marginal likelihood that is bounded and quite close to the full likelihood in information, as long as one is interested in the central part of the parameter space, away from its problematic boundaries. Our main goal is to show that the marginal likelihood solves the unboundedness problem in a manner competitive with other methods that were specifically designed for the normal mixture. To this end two algorithms have been developed. Their effectiveness is investigated through a simulation study. Finally, an application to real data is illustrated.