Mixtures of probability distributions and their application

This thesis is focused on finite mixtures of probability distributions. The primary goal is to acquaint readers with general and also deeper issues in this area. The whole text is divided into three parts. The first part is devoted to the theoretical description of selected methods. General principles of probabilistic mixtures, graphical methods for estimation of mixture parameters and estimation using EM algorithm are described. In addition, other possible methods for modeling with mixtures are outlined. The connection of the mixture models with the cluster analysis is also described. The second part, whose character is rather informative, focuses on the description of the data on which the selected methods are applied. The data come from the statistical survey Living conditions conducted by the Czech Statistical Office. The analyzed indicator is the annual disposable household income. The number of household members serves as an auxiliary variable. The last part of the thesis is focused on practical application. First of all, the basic principle of mixtures is described on the generated data by means of illustrations. Described is also a method of generating data from probability mixtures. Finally, the data described in the second part are analyzed. Estimates of the mixture parameters describing the annual disposable household income are made using graphical methods, EM algorithm and MLE. The best results were obtained using the EM algorithm. This algorithm is also applied in the presence of an auxiliary variable describing the number of household members. This model has also produced a good result.