Inference for Multivariate Normal Mixtures

TL;DR

This paper introduces a penalized likelihood approach for estimating multivariate normal mixture models, addressing issues of inconsistency in classical methods and providing an effective EM-algorithm with practical guidelines.

Contribution

It proposes a novel penalized likelihood estimator for multivariate normal mixtures and demonstrates its strong consistency and practical computation methods.

Findings

Penalized likelihood estimator is strongly consistent with known component bounds.

The EM-algorithm effectively computes the estimator in practice.

Simulation studies validate the method's effectiveness and limitations.

Abstract

Multivariate normal mixtures provide a flexible model for high-dimensional data. They are widely used in statistical genetics, statistical finance, and other disciplines. Due to the unboundedness of the likelihood function, classical likelihood-based methods, which may have nice practical properties, are inconsistent. In this paper, we recommend a penalized likelihood method for estimating the mixing distribution. We show that the maximum penalized likelihood estimator is strongly consistent when the number of components has a known upper bound. We also explore a convenient EM-algorithm for computing the maximum penalized likelihood estimator. Extensive simulations are conducted to explore the effectiveness and the practical limitations of both the new method and the ratified maximum likelihood estimators. Guidelines are provided based on the simulation results.