Asymptotic Theory For Generalized Linear Models

As a extension of classical linear models, Generalized Linear Models(GLM), which can make up the drawback of the linear models, has came into its popular application, Being suitable for both continuous and discrete data, especially the latter, such as counting data and categorical data. As well as the flexibility for different distribution and link function, GLM play an important role in biological, medical, agriculture, astronomy, economic and social data’s statistical analysis. More and more attractions and participations have been given into the study of the wide use of GLM, and the related theory about GLM has been expanded and developed with remorseless efforts by many experts and scholars.The asymptotic theory for generalized linear models is further discussed in this article. Firstly, with the assumptions that errors in ei=yi-Eyi,i=1,2,… is uncorrelated, and the conditions of with some smoothness conditions are true, it is proved that quasi-likelihood estimator βn of regression parameter vector p is weakly consistent with‖β-β0‖=Op((λmin∑i=1nZiZi’)1-/2),wnere yi and Zi are the function of response variables and covariate, λminA is the smallest characteristic root of A, β0is the true value of β. Secondly, Under the condition of{Zi,i≥1}is unbounded, and the conditions of‖ZAn‖=o(logn) and λmin∑i=1ZiZi’≥cnα are true, it is proved that the maximum likelihood estimation of the log linear gamma model is strong consistent and asymptotically normal.