The Asymptotic Theories For Generalized Estimating Equations

Generalized linear models are important extensions of the classical linear models. Generalized estimating equations method is which uses generalized linear model to analyze longitudinal data, and generalized estimating equations are created. Asymptotic theories of generalized estimating equations(GEE) can be regarded as extensions of generalized linear model theories, and can be widely applied in many fields, especially when dealing with discrete data, and now widely be used in medical, biology, agriculture, etc. The important references of this thesis have had an in-depth study of the asymptotic theories of one-dimension generalized estimating equations, and have given some corresponding conditions of asymptotic normality, weak consistency, strong consistency of generalized estimating equations, and these conditions are easier to test for several common models, such as the Logistic models, Probit models, group Cox models. This thesis mainly studies the asymptotic properties of multi-dimension generalized estimating equations’estimator, the main work of this paper includes:use fixed point theorem to prove the existence and asymptotic normality of generalized estimating equations’estimator when the link function is natural link function and residual is martingale sequence; prove strong consistency of the estimator respectively in three different circumstances:(1)n is limited, m is tend to be infinite (2)n is tend to be infinite, m is limited (3)n and m is tend to be infinite. Here n is observation number and m is the observation times for each individual. And prove the asymptotic existence and weak consistency of the root of pseudo likelihood equation with nature contact function.