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October 2009 Hypothesis test for normal mixture models: The EM approach
Jiahua Chen, Pengfei Li
Ann. Statist. 37(5A): 2523-2542 (October 2009). DOI: 10.1214/08-AOS651


Normal mixture distributions are arguably the most important mixture models, and also the most technically challenging. The likelihood function of the normal mixture model is unbounded based on a set of random samples, unless an artificial bound is placed on its component variance parameter. Moreover, the model is not strongly identifiable so it is hard to differentiate between over dispersion caused by the presence of a mixture and that caused by a large variance, and it 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 0.5χ02+0.5χ12 and χ12 distributions when the mixing variances are equal but unknown and the χ22 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.


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Jiahua Chen. Pengfei Li. "Hypothesis test for normal mixture models: The EM approach." Ann. Statist. 37 (5A) 2523 - 2542, October 2009.


Published: October 2009
First available in Project Euclid: 15 July 2009

zbMATH: 1173.62007
MathSciNet: MR2543701
Digital Object Identifier: 10.1214/08-AOS651

Primary: 62F03
Secondary: 62F05

Keywords: Chi-square limiting distribution , compactness , homogeneity test , likelihood ratio test , normal mixture models , statistical genetics

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 5A • October 2009
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