| Hu and Zidek (1997) proposed a very general method for using all relevant information in statistical inference. We extend their method to generalized linear models when the covariates are generated from different populations. Furthermore, we propose a unified and effective method to choose the weights using the estimated probability of membership through the expectation-maximization algorithm when memberships are unknown. The proposed weights are applicable to both discrete and continuous covariates and work well in generalized linear models. We also derive the asymptotic properties of the estimator based on the weighted likelihood equations.;Simulation studies are provided to demonstrate the performance of the new methods. Examples of the real data analysis are also presented. Our simulation studies suggested that the weighted likelihood with the proposed weights is more powerful in testing than the classical MLE. The weighted M-estimator is not only efficient in estimating the parameters in the main cluster, but also robust to influential points. Applications to real data sets imply that our method can detect real relationships that the classical method failed to discover.;In the end of the thesis, we also consider the weighted likelihood when the memberships of observations are known.;Notice that the method we proposed above only control the model uncertainty on the covariates directions, in the second part of the thesis, we employ robust techniques, such as M-estimation, to control the possible model violation on the response direction. M-estimator is a broad class of estimators which are obtained by minimizing certain functions of the data which have robust properties. |
