## 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.

### MSC:

 62F03 Parametric hypothesis testing

Zbl 0701.62106
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