R code for EM-test in mixture model

The EM-test is a likelihood based method for testing the order of a finite mixture. Some ideas of the EM-test can be traced back to the modified likelihood ratio test (Chen et al., 2001 and 2004), but it has its own advantages. (a)The EM-test has simple limiting distributions under less regularity conditions and therefore is more widely applicable. (b) Its theoretical analysis is less challenging.

More details can be found in the reference. So far this website mainly provides the R functions for

1. Testing the order of a finite mixture with univariate mixing parameter.

The EM-test can be used to  test

H_0: order of a mixture=m0 versus H_A: order of a mixture>m0,

where m0 is an arbitrary positive integer. The description and the properties of the EM-test can be found in Li and Chen (2009) and Li, Chen and Marriott (2009).

• R code for the EM-test under Binomial mixture: embinom.R
• R code for the EM-test under Exponential mixture: emexp.R
• R code for the EM-test under Normal mixture with known variance 1: emnorm.R
• R code for the EM-test under Poisson mixture: empois.R
• Readme file and some examples for the above R code: readme-emorder.pdf.
2. Testing the order of a normal mixtures with unknown variances.

Normal mixture distributions are arguably the most important mixture models, and also the most technically challenging. (a)The log-likelihood function is unbounded. (b) The Fisher information with respect to the mixing proportion maybe infinite. (c)The normal mixture model is not strongly identifiable if the variance parameter is unknown.

Our EM-test can easily handle the above three challenges. The description and properties can be found in Chen and Li (2010) and Chen, Li and Fu (2012).

The following are the R code which can be used for testing the order of a finite mixture.

• R code for testing the order of normal mixture with unknown variance: EMNormal2.R
If you use the R code above, please refer to the papers below where the methodology and properties are described.

Reference:

1. Chen, H., Chen, J., and Kalbﬂeisch, J. D. (2001)
A modiﬁed likelihood ratio test for homogeneity in ﬁnite mixture models. JRSSB, 63, 19-29.

2. Chen, H., Chen, J., and Kalbﬂeisch, J. D. (2004)
Testing for a ﬁnite mixture model with two components. JRSSB, 66, 95-115.

3. Chen, J. and Li, P. (2009).
Hypothesis test for normal mixture models: the EM approach. The Annals of Statistics, 37, 2523-2542.

4. Li, P. and Chen, J. (2010).
Testing the order of a finite mixture model. JASA, 105, 1084-1092.

5. Li, P., Chen, J. and Marriott, P. (2009).
Non-finite Fisher information and homogeneity: the EM approach. Biometrika, 96, 411-42.

6. Chen, J., Li, P.  and Fu, Y. (2012).
Inference on the order of a normal mixture. JASA, 107, 1096-1105.