Tests for Homogeneity in Normal Mixtures in the Presence of a Structural Parameter: Technical Details

Often a question arises as to whether the observed data are a sample from a homogeneous population or the data have come from a heterogeneous population. In particular, one wants to test for a single normal distribution versus a mixture of two normal distributions. Classic asymptotic results fail to apply to the problem since the model does not satisfy the regularity conditions. This paper investigates the large sample behavior of the likelihood ratio statistic for testing homogeneity in the normal mixture in location parameters with an unknown structural parameter. It is proved that the asymptotic null distribution of the likelihood ratio statistic is the maximum of a χ2-variable and supremum of the square of a truncated Gaussian process with mean 0 and variance 1. This reveals and exposes the unusual large sample behavior of the likelihood function under the null distribution. The correlation structure of the process involved in the limiting distribution is presented explicitly. From the large sample study, it is also found that even though the structural parameter is not part of the mixing distribution, the convergence rate of its maximum likelihood estimate is n−1/4 rather than n−1/2, while the mixing distribution has a convergence rate n−1/8 rather than n−1/4. This is in sharp contrast to the ordinary semi-parametric models and to the mixture models without a structural parameter.