Consistency And Asymptotic Normality Of Quasi-Estimator In Generalized Linear Models With Missing Data

The generalized linear model (GLM) which is suitable for the continuous data and the discrete data has great application in biology, medicine, economy and society field, etc. The objective of this paper is to discuss the consistency and asymptotic normality of Quasi-likelihood estimation under some regularity conditions when the covariates are incomplete.Firstly, based on the EM algorithm, a new mothed is put forward when the covariates and the response variables are both discrete and the covariate variables(column vectors) are missing at random. The new method which use the valid data linear predict the incomplete covariable data. Then consistency and asymptotic normality of the parameter estimation are proved under some suitable regularity conditions with this new method when the relationship is specified between the mean and variance of the Quasi-likelihood equation.Secondly, the new method is extended to the more general case by extending the column vector of covariates to the matrix, response variables are multidimensional and replacing the function relationship between the mean and variance by appropriate matrix value function. Then under some suitable regularity conditions, the strong consistency of the parameter Quasi-likelihood estimation is proved by the new method.Finally, some of the more common GLM models and test methods commonly used are introduced. Through the specific data simulation, the parameter estimations of cumulative odds model and the logistic model are compared. At last, the paper illustrate that when the classification is appropriate, the analyses of the two models are very similar. But if the classification has a more significant difference, the analysis result of cumulative logistic model is more reliable than logistic model.