Asymptotic Properties For Pseudo-Likelihood Equations

The general linear model is mainly used to deal with the continuous data of the response variables, with the requirement that the error term complies with normal distribution. The generalized linear model is the generalization of the general linear model, and it can be both used to deal with the continuous data and the discrete data of the response variables. Using the generalized linear model to analyze the longitudinal data is called the way of generalized estimating equation, and then generalized estimating equation is created. The real correlation matrix Ri of the generalized estimating equation is usually unknown, and is usually replaced by the working correlation matrixRi(a). The working correlation matrix Ri,(α) is chosen by the researchers according to their experience, in order to carry on the study. Through the samples, Balan and Schiopu got an estimator Rπ, which was used to replace Ri(a), to estimate the real correlation matrix Ri, and the resulting equation was called pseudo-likelihood equation. Besides, they brought about the conditions of weak consistency and asymptotic normality of the radicals of the pseudo-likelihood equation under the circumstances of natural correlation and limited repeated observation with martingale difference sequence as its residuals. Their methods made the estimation more objective and reduced the artificial factors of Ri (α). Basing on their outcomes, this thesis further studies the theory of the pseudo-likelihood equation. The major work of thesis include:(i) a new working correlation matrix is given out and a new pseudo-likelihood equation is introduced;(ii) it proves that under the circumstance of natural correlation, even if the repeated observation is limited and the residuals are martingale difference sequence, the new pseudo-likelihood equation still has the properties of weak consistency and asymptotic normality, thus Balan and Schiopu’s study is further developed;(iii) it simulates and examines the theoretical results with numbers, and proves the advantage of the estimation method.