Using Finite Mixtures to Robustify Statistical Models

Abstract

Abstract This thesis is concerned with robust estimation of the parameters of statistical models. Although robust estimation is a very good idea, it has some shortcomings when seen from the statistical modelling point of view. For example, there are no easily applicable principles for creating robust estimates in new situations. In this thesis, we are trying to introduce a unified method to obtain the robustified statistics for various situations such as the linear model and the generalized linear model. We wish to modify the maximum likelihood estimation procedure, which is very sensitive to the outliers. In order to reduce the effect on these estimates by outliers, we add an additional component, which would be of no interest but would contain all outliers, to the regular component forming a finite mixture. In fact, we use the finite mixture model to obtain a ``robustified'' estimate for a model parameter $\theta$, where the finite mixture form is being used as a mathematical tool to have a tractable form of analysis rather than being used as a serious model for the data. We employ the EM algorithm to obtain the our proposed robustified estimates for the parameters. Our estimates are compared with some other estimates defined in the robust statistics literature. This thesis examines the robustness of the proposed estimates using the concept of influence function. The estimates are defined iteratively, so that the implicit differentiation method of Jorgensen is used to obtain the influence functions of the estimates. We give example plots of these influence functions which are bounded. In this thesis, we give mathematical results for all cases and we use well known real data sets to investigate our method. The statistical software R is used for all investigation. Finally, we hope that this method may give a unified approach for making parameter estimation in statistical models more robust.