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Biometrika 2001 88(3):767-778; doi:10.1093/biomet/88.3.767
© 2001 by Biometrika Trust
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Testing the number of components in a normal mixture

Yungtai Lo1, Nancy R.Mendell2 and Donald B.Rubin3

1 Department of Statistics, Harvard University, Cambridge, Massachusetts 02138, U.S.Aytlo{at}hustat.harvard.edu 2 Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, New York 11794, U.S.A.nmendell{at}notes.cc.sunysb.edu 3 Department of Statistics, Harvard University, Cambridge, Massachusetts 02138, U.S.A. rubin{at}hustat.harvard.edu

We demonstrate that, under a theorem proposed by Vuong, the likelihood ratio statistic based on the Kullback–Leibler information criterion of the null hypothesis that a random sample is drawn from a k0-component normal mixture distribution against the alternative hypothesis that the sample is drawn from a k1-component normal mixture distribution is asymptotically distributed as a weighted sum of independent chi-squared random variables with one degree of freedom, under general regularity conditions. We report simulation studies of two cases where we are testing a single normal versus a two-component normal mixture and a two-component normal mixture versus a three-component normal mixture.An empirical adjustment to the likelihood ratio statistic is proposed that appears to improve the rate of convergence to the limiting distribution.

Key Words: Kullback–Leibler information criterion; Likelihood ratio test; Normal mixture; Weighted sum of chi-squared random variables

Received April 2000. Revised January 2001


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