### Abstract

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 language | English (US) |
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Pages (from-to) | 767-778 |

Number of pages | 12 |

Journal | Biometrika |

Volume | 88 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2001 |

Externally published | Yes |

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### Keywords

- 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

### Cite this

*Biometrika*,

*88*(3), 767-778. https://doi.org/10.1093/biomet/88.3.767