This research on mixture models will develop graphical diagnostics and formal tests to identify the presence of mixing in generalized linear models and to determine the number of mixing components for finite mixtures. Methods will also be developed to incorporate full information for case-control studies with errors in variables. The theory will draw on ideas and techniques from nonparametric maximum likelihood, empirical processes, asymptotics, convex geometry, total positivity and from simulation. Mixture distributions arise when homogeneous populations are combined in such a way that the origins of individuals are lost. A fundamental problem involves determining whether or not this combining has occurred and if so, how many different originating populations there were. This research will investigate graphical techniques and will use mathematical theory to develop practical statistical methods to answer these kinds of questions in many scientific applications.
