Consistency And Asymptotic Normality Of Maximum Quasi-likelihood Estimators In Generalized Linear Models

The generalized linear model, which is the classical linear model promotion, is a kind of very important mathematical model and has been widely used. It is very significant in data analysis in the economy, the society, the medicine, in biology and so on. It is suitable for the continuous and the discrete data, particularly the latter, like counted data, characteristic data and so on. The generalized linear model includes models such as the peculiar circumstance, the linear regression, the variance analysis model, the logarithm and the probit model of alternated responds, the log-linear model, the counting many response model and some commonly used models of survival data. There are massive properties in some models above, such as the linearity. We can use these properties to obtain the very good effect. In addition, we also have commonly used methods of the parameter estimation.Generalized linear model's example originated very early. World-famous statistician Fish once used this model in 1919. In 1972 Nelder and Wedderburn introduced the concept of the generalized linear model in a paper. In 1989 McCullagh and Nelder discussed the generalized linear model and obtained achievement in their reprinted work in detail. Now there is much literature in this aspect.In this paper, parameter estimation of the generalized linear model is studied. The asymptotic properties of parameter estimation are discussed, including consistency, asymptotic existence and normality.1. The consistency of maximum quasi-likelihood estimators (MQLEs) in generalized linear models with natural link function and adaptive designs is discussed. When the response yi is q x 1 dimensional random vectors, the p x q regressors Xi is bounded, the minimum eigenvalue sup E and the other mild conditions, the consistency, asymptotic existence and the rate of MQLEs are proved. Corresponding results are not obtained until now when2. The asymptotic normality of maximum quasi-likelihood estimators (MQLEs) in generalized linear models with natural link function and adaptive designs is discussed. When the response yi is q x 1 dimensional random vectors, the p x q regressors Xi is bounded, and the other mild conditions, the asymptotic normality of MQLEs is proved. This weakened in Gao Q B and Wu Y H (2004).