Hypothesis test for normal mixture models: The EM approach
By Jiahua Chen, Pengfei Li. 2009
192 citations recorded by the source corpus.
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 $0.5\chi^2_0+0.5\chi^2_1$ and $\chi^2_1$ 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.
Subfields: Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/c6903d2703142a82228fcc8baad0e02c5e7da646
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Artificial Intelligence
- Michael I. Jordan
- Hiroyuki Kasahara
- Katsumi Shimotsu
- Martin J. Wainwright
- Bin Yu
- Pengfei Li
- David M. Blei
- Yuejiao Fu
In Statistics and Probability
- Martin J. Wainwright
- Michael I. Jordan
- Hiroyuki Kasahara
- Katsumi Shimotsu
- Bin Yu
- Pengfei Li
- Francis R. Bach
- Kengo Kato
Citations in the corpus
- Inference on the Order of a Normal Mixture (2012, methodology): “Second, because the Fisher information with respect to the mixing proportion can be infinity (Chen and Li 2009), the limiting distribution may not even exist.”
- Mixture of Regression Models With Varying Mixing Proportions: A Semiparametric Approach (2012, background): “For other references, see McLachlan and Peel (2000); Chen, Chen, and Kalbfleisch (2004); and Chen and Li (2009).”
- Inference for Subgroup Analysis With a Structured Logistic-Normal Mixture Model (2015, background): “Recently, Chen and Li (2009) and Li and Chen (2010) proposed the so-called EM test that does not rely on the maximized likelihood.”
- Testing the Number of Components in Normal Mixture Regression Models (2015, methodology): “In addition to the failure of regularity conditions that is common to all finite mixture models, the normal mixture model (2) has additional undesirable mathematical properties, as discussed in Chen and Li (2009): (a) The Fisher information for testing H02 is not finite unless the range of σ 2 1 /σ ”
- Singularity, misspecification and the convergence rate of EM (2018, background): “This problem is known to be challenging due to the break-down of the (generalized) likelihood ratio test around the singularity of the Fisher information matrix; see the papers [19, 5] for some past work on the problem.”
- Testing the Order of a Finite Mixture (2010, methodology): “To develop a method without such restrictive conditions on the mixture models, Li, Chen, and Marriott (2009) and Chen and Li (2009) have proposed an EM test.”