Testing the number of components in a normal mixture

Yungtai Lo, Nancy R. Mendell, Donald B. Rubin

Research output: Contribution to journalArticlepeer-review

3199 Scopus citations


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 A-0-component normal mixture distribution against the alternative hypothesis that the sample is drawn from a A-4-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.

Original languageEnglish (US)
Pages (from-to)767-778
Number of pages12
Issue number3
StatePublished - 2001
Externally publishedYes


  • Kullback-Leibler information criterion
  • Likelihood ratio test
  • Normal mixture
  • Weighted sum of chi-squared random variables

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Agricultural and Biological Sciences (miscellaneous)
  • Agricultural and Biological Sciences(all)
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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