| The main purpose of this thesis is to introduce methodology to fit finite mixture models. We propose a method where the component distributions are fitted using an exponential tilt model, in which the log ratio of the density functions of the components is modeled as a quadratic function of the observations. This approach has two key advantages. First, except for the exponential tilt assumption, the marginal distributions of the observations can be completely arbitrary. Second, we have the advantage of having a likelihood. In the multivariate case, a model selection method is introduced for identifying the number of component distributions in the mixture model when theory does not suggest it. In addition, a likelihood ratio test, using the profile likelihood, is used to choose between a model that assumes the coordinates are conditionally independent and one that has blocks of conditionally i.i.d. coordinates. Simulations are provided to show the performance of the methods in estimating the component means and standard deviations. The proposed methods are applied to real datasets. |
