### Abstract

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ^{2} reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ^{2} reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795-800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.

Original language | English (US) |
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Pages (from-to) | 233-240 |

Number of pages | 8 |

Journal | Statistics and Computing |

Volume | 18 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2008 |

Externally published | Yes |

### Fingerprint

### Keywords

- Bootstrap
- EM algorithm
- Likelihood ratio test
- Normal mixture

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Statistics and Probability

### Cite this

**A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture.** / Lo, Yungtai.

Research output: Contribution to journal › Article

*Statistics and Computing*, vol. 18, no. 3, pp. 233-240. https://doi.org/10.1007/s11222-008-9052-4

}

TY - JOUR

T1 - A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

AU - Lo, Yungtai

PY - 2008/9

Y1 - 2008/9

N2 - It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ2 reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ2 reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795-800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.

AB - It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ2 reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ2 reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795-800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.

KW - Bootstrap

KW - EM algorithm

KW - Likelihood ratio test

KW - Normal mixture

UR - http://www.scopus.com/inward/record.url?scp=45449093770&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45449093770&partnerID=8YFLogxK

U2 - 10.1007/s11222-008-9052-4

DO - 10.1007/s11222-008-9052-4

M3 - Article

AN - SCOPUS:45449093770

VL - 18

SP - 233

EP - 240

JO - Statistics and Computing

JF - Statistics and Computing

SN - 0960-3174

IS - 3

ER -