On Bootstrapping the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture

SUMMARY An important but difficult problem in practice is assessing the number of components g in a mixture. An obvious way of proceeding is to use the likelihood ratio test statistic λ to test for the smallest value of g consistent with the data. Unfortunately with mixture models, regularity conditions do not hold for –2 log λ to have it usual asymptotic null distribution of chi‐squared. In this paper the role of the bootstrap is highlighted for the assessment of the null distribution of –2 log λ for the test of a single normal density versus a mixture of two normal densities in the univariate case.