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Title: A classical invariance approach to the normal mixture problem Authors:  Monia Ranalli - The Pennsylvania State University (United States) [presenting]
Bruce Lindsay - The Pennsylvania State University (United States)
David Hunter - Pennsylvania State University (United States)
Abstract: 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 to the likelihood. In the univariate case there are $n$ solutions, corresponding to the $n$ distinct data points, along a parameter boundary, each with an infinite spike of the likelihood, none making particular sense as a chosen solution. In the multivariate case, there is an even more complex likelihood surface. 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 and in a manner competitive with other methods that were specifically designed for the normal mixture. To this aim, two different algorithms have been developed. Their effectiveness is investigated through a simulation study. Finally, an application to real data is illustrated.