Testing the number of components in a normal mixture. (English) Zbl 0985.62019

Summary: We demonstrate that, under a theorem proposed by Q.H. Vuong [Econometrica 57, No. 2, 307-333 (1989; Zbl 0701.62106)], the likelihood ratio statistic based on the Kullback-Leibler information criterion or the null hypothesis that a random sample is drawn from a \(k_0\)-component normal mixture distribution against the alternative hypothesis that the sample is drawn from a \(k_1\)-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.


62F03 Parametric hypothesis testing


Zbl 0701.62106
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