Penalized Empirical Likelihood Method Based On Generalized Estimating Equations For Longitudinal Data

Longitudinal data with repeated observations are often generated in biomedical,clinical trials,economic and financial fields.How to effectively analyze such data is one of the hot issues concerned by statistical practitioners and researchers.Generalized estimation equation method is a special method to deal with longitudinal data with repeated observations.Empirical likelihood is a very effective nonparametric statistical inference method.Compared with the classical or modern statistical methods,empirical likelihood has many outstanding advantages,such as the confidence interval constructed has the domain retention and transformation invariance,the shape of the confidence region is determined by the data itself,Bartlett correction and no need to construct the axis statistics,etc.The penalty method is an effective statistical inference method developed in recent years.It is widely used in variable selection,parameter estimation and hypothesis testing of models.In this paper,we consider the theory and application of penalized empirical likelihood method for longitudinal data based on generalized estimation equation.In this paper,we discuss the parameter estimation,variable selection and hypothesis testing of balanced longitudinal data under the penalized empirical likelihood method with generalized estimation equation as auxiliary information.Firstly,we construct the generalized estimation equation by using the estimated correlation matrix.Under appropriate assumptions,we establish the existence?consistency?Oracle property of the penalized empirical likelihood estimator,and the asymptotic chi-square property of the penalized empirical likelihood ratio test statistic under null hypothesis based on the generalized estimation equation.Then,we demonstrate the validity of the proposed method through numerical simulation: in the simulation of parameter estimation accuracy,the proposed method is compared with the generalized estimation equation method?empirical likelihood method and penalized generalized estimation equation method,and the simulation results show that the accuracy of the proposed method is higher than the other three methods;in the simulation of the validity of variable selection,our method is compared with the penalized generalized estimation equation method,and the result of variable selection of our method is better than that of the penalized generalized estimation equation method;in addition,the size and power of the penalized empirical likelihood ratio test are simulated,and the results are very satisfactory.Finally,we use the proposed method to analyze the economic income data of the United States.Compared with the existing research literature,the feature of this paper is using the data to estimate the correlation structure.On this basis,we establish the asymptotic results of parameter estimation,variable selection and hypothesis testing of the penalized empirical likelihood method.This method can get rid of the subjectivity and arbitrariness of specifying the correlation structure in advance,and ensure the validity of statistical inference.This is also the innovation of this paper.