Inference on the Order of a Normal Mixture

Abstract

Finite normal mixture models are used in a wide range of applications. Hypothesis testing on the order of the normal mixture is an important yet unsolved problem. Existing procedures often lack a rigorous theoretical foundation. Many are also hard to implement numerically. In this article, we develop a new method to fill the void in this important area. An effective expectation-maximization (EM) test is invented for testing the null hypothesis of arbitrary order m ₀ under a finite normal mixture model. For any positive integer m ₀ ⩾ 2, the limiting distribution of the proposed test statistic is . We also use a novel computer experiment to provide empirical formulas for the tuning parameter selection. The finite sample performance of the test is examined through simulation studies. Real-data examples are provided. The procedure has been implemented in R code. The p-values for testing the null order of m ₀ = 2 or m ₀ = 3 can be calculated with a single command. This article has supplementary materials available online.

KEY WORDS:

• Chi-squared limiting distribution
• Computer experiment
• EM test
• Likelihood ratio test
• Order selection
• Tuning parameter
• Unequal variance

Acknowledgments

The authors thank the editor, the associate editor, and three referees for constructive comments and suggestions that lead to significant improvements in the article. The research is supported by the Natural Sciences and Engineering Research Council of Canada and by a start-up grant from the University of Waterloo.