A note on estimating multivariate Gaussian mixtures with unknown number of components

In this article, a new penalized likelihood method is proposed for finite multivariate Gaussian mixture models to conduct order selection and model estimation. The method is proved to achieve order selection consistency and have root-n convergence rate. Asymptotic normality is established for the proposed estimator. A modified EM algorithm is developed for computation. Extensive simulations and a real data analysis are conducted to illustrate the performance of our method.