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**Hypothesis test for normal mixture models: the EM approach.**
*(English)*
Zbl 1173.62007

Summary: Normal mixture distributions are arguably the most important mixture models, and also the most technically challenging. The likelihood function of normal mixture models is unbounded based on a set of random samples, unless an artificial bound is placed on its components variance parameters. Moreover, the model is not strongly identifiable so it is hard to differentiate between overdispersion caused by the presence of a mixture and that caused by a large variance, and has infinite Fisher information with respect to mixing proportions. There has been extensive research on finite normal mixture models, but much of it addresses merely consistency of the point estimation or useful practical procedures, and many results require undesirable restrictions on the parameter space.

We show that an EM-test for homogeneity is effective at overcoming many challenges in the context of finite normal mixtures. We find that the limiting distribution of the EM-test is a simple function of the \(0.5\chi _{0}^{2}+0.5\chi _{1}^{2}\) and \(\chi _{1}^{2}\) distributions when the mixing variances are equal but unknown and the \(\chi _{2}^{2}\) when variances are unequal and unknown. Simulations show that the limiting distributions approximate the finite sample distribution satisfactorily. Two genetic examples are used to illustrate the application of the EM-test.

We show that an EM-test for homogeneity is effective at overcoming many challenges in the context of finite normal mixtures. We find that the limiting distribution of the EM-test is a simple function of the \(0.5\chi _{0}^{2}+0.5\chi _{1}^{2}\) and \(\chi _{1}^{2}\) distributions when the mixing variances are equal but unknown and the \(\chi _{2}^{2}\) when variances are unequal and unknown. Simulations show that the limiting distributions approximate the finite sample distribution satisfactorily. Two genetic examples are used to illustrate the application of the EM-test.

### MSC:

62F03 | Parametric hypothesis testing |

65C60 | Computational problems in statistics (MSC2010) |

62E20 | Asymptotic distribution theory in statistics |

62F05 | Asymptotic properties of parametric tests |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

### Keywords:

chi-square limiting distribution; compactness; normal mixture models; homogeneity test; likelihood ratio test; statistical genetics
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\textit{J. Chen} and \textit{P. Li}, Ann. Stat. 37, No. 5A, 2523--2542 (2009; Zbl 1173.62007)

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