Degeneracy in the Maximum Likelihood Estimation of Univariate Gaussian Mixtures for Grouped Data and Behaviour of the EM Algorithm
Christophe Biernacki · Scandinavian Journal of Statistics · 2007
Abstract. In the context of the univariate Gaussian mixture with grouped data, it is shown that the global maximum of the likelihood may correspond to a situation where a Dirac lies in any non‐empty interval. Existence of a domain of attraction near such a maximizer is discussed and we establish that the expectation‐maximization (EM) iterates move extremely slowly inside this domain. These theoretical results are illustrated both by some Monte‐Carlo experiments and by a real data set. To help practitioners identify and discard these potentially dangerous degenerate maximizers, a specific stopping rule for EM is proposed.