Asymptotic Properties Of The Generalized Estimation Equation (GEE) For High Dimensional Longitudinal Categorical Data

The generalized linear model(GLM)is a generalization of the common linear regression model.All of the models are belong to the category of the study of generalized linear models that the density function of the variables can express as the exponential distribution.For example,the common normal distribution,Poisson distribution and two point distribution can be considered as a generalized linear model.It can be applied to continuous data and discrete data,especially discrete data,such as categorical data and counting data.Therefore,the generalized linear model is widely applicable.Longitudinal data is the data that each individual is repeated measurements.The observed values of different individuals are independent.Besides,the observed values for the same individual is relevant.Longitudinal data is more common in economics,sociology,medical research.Due to the application widely of the generalized linear models,a large number of experts and scholars have attracted to research the generalized linear models,which makes the theory of the generalized linear models more perfect,and have made some achievements in practical applications.At the same time,with the arrival of the large data age,the sample size n tends to infinity,and the covariate dimension p will tend to infinity.However,most experts and scholars study the case that the sample size n tends to infinity and the covariate dimension p is fixed(finite)at present.Therefore,we analyse the case that the covariate dimensionality is divergence.It is very important to study the longitudinal data.In this paper,we will reseach the the two stage of Logit model.The sample size n will tend to infinity and the covariate dimension is divergence.Under the mild conditions,according to the existence theorem of the roots of the nonlinear multivariable equations,the central limit theorem and the Cauchy-Schwarz inequality,we proved the existence,consistency and asymptotic normality of the estimation of generalized estimating equation.On the other hand,we can generalize the response variable from one-dimensional to two-dimensional and extend the natural link function to the general link function.