Estimation of the Complexity of a Finite Mixture Distribution: From Well- to Less Known Methods

Abstract

Mixture models occur in numerous settings including random and fixed effects models, clustering, deconvolution, empirical Bayes problems and many others. They are often used to model data originating from a heterogeneous population, consisting of several homogeneous subpopulations, and the problem of finding a good estimator for the number of components in the mixture arises naturally. Estimation of the order of a finite mixture model is a hard statistical task, and multiple techniques have been suggested for solving it. We will concentrate on several methods that have not gained much popularity yet deserve the attention of practitioners. These can be categorized into three groups: tools built upon the determinant of the Hankel matrix of moments of the mixing distribution, minimum distance estimators, likelihood ratio tests. We will address theoretical pillars underlying each of the methods, provide some useful modifications for enhancing their performance and present the results of the comparative numerical study that has been conducted under various scenarios. According to the results, none of the methods proves to be a “magic pill”. The results uncover limitations of the techniques and provide practical hints for choosing the best-suited tool under specific conditions.