### 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) |
---|---|

Pages (from-to) | 767-778 |

Number of pages | 12 |

Journal | Biometrika |

Volume | 88 |

Issue number | 3 |

State | Published - 2001 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Agricultural and Biological Sciences(all)
- Agricultural and Biological Sciences (miscellaneous)
- Statistics and Probability
- Mathematics(all)
- Applied Mathematics

### Cite this

*Biometrika*,

*88*(3), 767-778.

**Testing the number of components in a normal mixture.** / Lo, Yungtai; Mendell, Nancy R.; Rubin, Donald B.

Research output: Contribution to journal › Article

*Biometrika*, vol. 88, no. 3, pp. 767-778.

}

TY - JOUR

T1 - Testing the number of components in a normal mixture

AU - Lo, Yungtai

AU - Mendell, Nancy R.

AU - Rubin, Donald B.

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - Kullback-Leibler information criterion

KW - Likelihood ratio test

KW - Normal mixture

KW - Weighted sum of chi-squared random variables

UR - http://www.scopus.com/inward/record.url?scp=0038183179&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038183179&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0038183179

VL - 88

SP - 767

EP - 778

JO - Biometrika

JF - Biometrika

SN - 0006-3444

IS - 3

ER -