Large sample distribution of the likelihood ratio test for normal mixtures

This article concerns with the problem of testing whether a mixture of two normal distributions with bounded means and specific variance is simply a pure normal. The large sample behavior of the likelihood ratio test for the problem is carefully investigated. In the case of one mean parameter, it is shown that the large sample null distribution of the likelihood ratio test statistic is the squared supremum of a Gaussian process with zero mean and explicitly given covariances. In the case of two mean parameters, both the simple and composite hypotheses of normality are considered. Under the simple null hypothesis, the large sample null distribution is found to be an independent sum of a chi-square variable and the squared supremum of another Gaussian process whose covariance structure is slightly different from the one mean parameter case, while under the composite null hypothesis, the chi-square term is absent.