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

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 A{ to test for the smallest value of g consistent with the data. Unfortunately with mixture models, regularity conditions do not hold for -2 log A, 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 A{ for the test of a single normal density versus a mixture of two normal densities in the univariate case.

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