A note on ‘Testing the number of components in a normal mixture’

In a recent paper, Lo et al.(2001) propose a test for the likelihood ratio statistic based on the Kullback–Leibler information criterion when testing the null hypothesis that a random sample is drawn from a mixture of k0 normal components against the alternative hypothesis of a mixture with k1 normal components with k0 less than k1. However, this result requires conditions that are generally not met when the null hypothesis holds. Consequently, the result is not proven and simulations suggest that it may not be correct.