Estimating the number of components in finite mixture models via the Group-Sort-Fuse procedure

Abstract

Estimation of the number of components (or order) of a finite mixture model is a long standing and challenging problem in statistics. We propose the Group-Sort-Fuse (GSF) procedure—a new penalized likelihood approach for simultaneous estimation of the order and mixing measure in multidimensional finite mixture models. Unlike methods which fit and compare mixtures with varying orders using criteria involving model complexity, our approach directly penalizes a continuous function of the model parameters. More specifically, given a conservative upper bound on the order, the GSF groups and sorts mixture component parameters to fuse those which are redundant. For a wide range of finite mixture models, we show that the GSF is consistent in estimating the true mixture order and achieves the ${\mathit{n}}^{-1/2}$ convergence rate for parameter estimation up to polylogarithmic factors. The GSF is implemented for several univariate and multivariate mixture models in the R package $\mathtt{GroupSortFuse}$. Its finite sample performance is supported by a thorough simulation study, and its application is illustrated on two real data examples.