Asymptotic Properties Of Penalized Generalized Estimating Equations For High-Dimensional Longitudinal Count Data

Generalized linear model(GLM)as an important extension of the classi-cal linear model is suitable not only for continuous data but also for discrete data, and the penalized generalized linear model(PGLM) is added with the penalty factor in the generalized linear model, it is also applicable to contin-uous data and discrete data, for example, counted data and categorical data. In this paper, the high dimensional longitudinal data is the longitudinal data which is obtained in the case of the infinite of the covariates pn. The gen-eralized estimating equation(GEE)introduced by Liang and Zeger(1986) for the first time in theory and in practice, they have been greatly developed, and have achieved fruitful results. And it is mainly applied to the regression anal-ysis of longitudinal data. Among them, Wang Lan et al.(Biometrics,2012: 353-360)in their article, the penalty factor takes into account the general-ized estimating equation, which produces a penalized generalized estimating equation(PGEE). Penalty factor is mainly used to control the regression co-efficients of generalized estimating equations, decided whether generalized estimating equations of some covariates should be excluded from the final model selection, so as to improve the accuracy and efficiency of the estima-tion and inference.The main aim at the paper is to prove the large sample theory, such as the existence, consistency and asymptotic normality, of PGEE in the Possion dis-tribution models with the canonical link functions via using Limit Theory of Probability, Cauchy-Schwarz Inequality, and the Markov’s Inequality, espe-cially the technique of existence theorem of multivariate nonlinear equations root, when the simple sizes nâ†'∞, and the dimension of covariates pnâ†'∞, under the assumption that conditions are established.