Inference on the Order of a Normal Mixture

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 chi(2)(2m0). 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.