Testing the number of components in a normal mixture.

We demonstrate that, under a theorem proposed by Vuong, the likelihood ratio statistic based on the Kullback–Leibler information criterion of the null hypothesis that a random sample is drawn from a k0‐component normal mixture distribution against the alternative hypothesis that the sample is drawn from a k1‐component normal mixture distribution is asymptotically distributed as a weighted sum of independent chi‐squared random variables with one degree of freedom, under general regularity conditions. We report simulation studies of two cases where we are testing a single normal versus a two‐component normal mixture and a two‐component normal mixture versus a three‐component normal mixture. An empirical adjustment to the likelihood ratio statistic is proposed that appears to improve the rate of convergence to the limiting distribution. [ABSTRACT FROM PUBLISHER]