UC Riverside

Finite normal mixture models are often used to model the data coming from a population which consists of several homogeneous sub-populations. The likelihood of normal mixture models with unequal variance is unbounded, and thus its maximum likelihood estimate (MLE) is not well defined. The likelihood equation of finite normal mixture model has multiple roots. Simply using the root with the largest likelihood will not always work because of the spurious roots which contain components with scales (or spreads) too different to be credible. In addition, General likelihood ratio tests (LRT) are not valid because of the violation in some conditions. The asymptotic distribution of the LRT breaks down due to unbounded of the likelihood function, and the model is not strongly identifiable.

This dissertation addresses the above two problems (root selection and goodness of fit for finite normal mixture models). For root selection, we consider using distance based on the goodness of fit statistics to choose the right root of maximum likelihood estimate. The results show that Cr$\acute{a}$mer Von Mises (CVM) statistic has overall the best performance. For the goodness of fit, we argue for using statistics based on the empirical distribution function (EDF) for testing hypothesis on finite normal mixture models. Bootstrap technique is used to simulate the asymptotic distribution of goodness of fit statistics under the null hypothesis. We also extend our idea to multivariate normal mixture models. Instead of testing a goodness of fit directly to multivariate data, we project data to one dimension by the direction vector $(\ba)$. The suggested procedure is simple and can be applied to other multivariate mixture models.