Multivariate normal mixtures: A fast consistent method of moments

A longstanding difficulty in multivariate statistics is identifying and evaluating nonnormal data structures in high dimensions with high statistical efficiency and low search effort. Here the possibilities of using sample moments to identify mixtures of multivariate normals are investigated. A particular system of moment equations is devised and then shown to be one that identifies the true mixing distribution, with some limitations (indicated in the text), and thus provides consistent estimates. Moreover, the estimates are shown to be quickly calculated in any dimension and to be highly efficient in the sense of being close to the values of the parameters that maximize the likelihood function. This is shown by simulation and the application of the method to Fisher’s iris data. While establishing these results, we discuss certain limitations associated with moment methods with regard to uniqueness and equivariance and explain how we addressed these problems.