Some Questions Researching For Generalized Linear Models

Generalized linear model, which is a widely used statistical model, is the natural extention of the classical linear model. It has been applied to many fields, such as medicine, biology, economy, education, military, etc. Liang and Zeger apply this model to analyze longitudinal data and classify the relevant data. Generalized Estimating Equations (GEE) is developed by it. GEE is developed from GLM and its large-sample properties are constantly enriched and perfected. Liang and Zeger apply the method of maximum quasi-likelihood estimator (MQLE) to prove that GEE has weak consistent solutions under the normal conditions. However, these solutions are based on the condition that the correct supposition of the construction of the moments and link function are needed. The large-sample properties of these solutions do not depend on the supposition of working correlation matrix, whether it is correct or wrong. Then Zeger and Liang point out that if the deletion of longitudinal data is random, the stability of GEE's solutions will remain. Based on the research findings of Fahrmeir and Kaufmann,Xie and Yang further perfect the asymptotic theory of GEE. However, it is done under more complex conditions. This thesis gives the conditions to the usual logistic regression models and probit regression models, which are much simpler than the ones given by Xie and Yang. Under these conditions, the weak consistency and asymptotic normality of GEE estimator are verified. At the same time, a fixed-point theorem is used to develop the weak consistency of GEE estimator. This thesis also proves the strong consistency of the GEE estimator for such a special situation:the individual number is 1, but the number of repeated observation tends to be infinit with its residual as martingale difference sequence. In the last chapter, the thesis adopts mathematical modeling to analyze the application of generalized linear model which is used in Non-life insurance actuarial.