A simple root selection method for univariate finite normal mixture models

Abstract–

It is well known that there exist multiple roots of the likelihood equations for finite normal mixture models. Selecting a consistent root for finite normal mixture models has long been a challenging problem. Simply using the root with the largest likelihood will not work because of the spurious roots. In addition, the likelihood of normal mixture models with unequal variance is unbounded and thus its maximum likelihood estimate (MLE) is not well defined. In this paper, we propose a simple root selection method for univariate normal mixture models by incorporating the idea of goodness of fit test. Our new method inherits both the consistency properties of distance estimators and the efficiency of the MLE. The new method is simple to use and its computation can be easily done using existing R packages for mixture models. In addition, the proposed root selection method is very general and can be also applied to other univariate mixture models. We demonstrate the effectiveness of the proposed method and compare it with some other existing methods through simulation studies and a real data application.

Keywords:

• Finite normal mixture
• goodness of fit
• maximum likelihood
• root selection
• spurious roots
• unbounded likelihood

Acknowledgement

The authors are grateful to the editor, the associate editor, and the referees for their insightful comments and suggestions, which greatly improved this article.

Additional information

Funding

Yao’s research is supported by NSF grant DMS-1461677 and Department of Energy with the award No: DE-EE0007328. Yang’s research was supported by the National Nature Science Foundation of China grant 11471086, the National Social Science Foundation of China grant 16BTJ032, the Fundamental Research Funds for the Central Universities 15JNQM019, the National Statistical Scientific Research Center Projects 2015LD02, Science and Technology Program of Guangzhou 2016201604030074, and Science and Technology Planning Project of Guangdong 2016A050503033.