Study On Several Models For Correlated Data Analysis

Based on the theory of generalized estimating equations(GEE), this thesisstudies the parameter estimation in several statistical models for longitudinal data.Firstly, the parameter estimation in a typical hierarchical generalized linearmodels is considered. Generalized linear models(GLM) include lots of practicalvalue models and it has a number of eximious properties, so that it is widelyapplied to various fields including biometrics, finance, insurance and so on. How-ever, in practical situations, the data come from longitudinal research or stratifieddesign structures, the sample with dependence or the character of overdispersionbe presented when we fitting the classical distributions, then we require addingrandom e?ects to the models and GLM is not fit for this circumstance. By in-cluding random components in the linear predictor with arbitrary distributionsin generalized linear models, Lee and Nelder introduced hierarchical generalizedlinear models(HGLM). At the same time, a new framework of estimation, whichwe call L-N method, was proposed. However, it needs a further theoretical investi-gation. We first study the L-N estimators in Poisson-Gamma(P-G) models whichare typical hierarchical generalized linear models for longitudinal data along thesimilar lines of GEE. On the one hand, under proper assumptions on responsevariables and some smoothing conditions, we obtain the strong consistency andthe convergence rate along with the asymptotic normality of the L-N estimatorsfor the fixed e?ectβin P-G models. On the other hand, the L-N method is proved pretty good by simulations in the cases of small and moderate sample sizes. Fur-thermore, Kolmogorov test show that the L-N method works well for the randome?ects predications. Moreover, we discuss the hypothesis testing of parameterbased on the theoretical results for the large sample.Secondly, the heteroscedasticity among di?erent individuals in the same trialgroup is not only behaved by covariables in practical situations. Then, we extendPoisson-Gamma models to the extended Poisson-Gamma modes, i.e. we are notonly put the random e?ect within group to the models, but also we add the in-dividual random e?ect to the models, moreover, other HGLM can be extendedby the same way, and thereby HGLM be generalized. And we first study it byusing L-N method. In an extended Poisson-Gamma model family, theoretical re-search shows that the L-N estimators have the analogous large sample propertiesto MLE; Monte Carlo Simulation method is employed to compare the L-N estima-tor and maximum marginal likelihood estimator for small and moderate samplesizes. It is shown that the L-N estimator appears much better than the maximummarginal likelihood estimator, which is di?cult to implement because of the mul-tiple integration involved. The models proposed here could be used to account foroverdispersion, heteroscedasticity, and correlation among repeated observations.Furthermore, we use it to analysis the epileptic seizure count data arising froma study of progabide as an adjuvant therapy for partial seizures, the results aresatisfying.Thirdly, base on above research works, we also discuss the parameter es-timation in reproductive dispersion models in this paper. About reproductive dispersion models(RDM), Bo-Cheng Wei and Nian-Sheng Tang et al investigatednonlinear reproductive dispersion models which is a type of RDM. Furthermore,nonlinear reproductive dispersion random e?ects models, which is also a kind ofRDM, are studied by Wen-Zhuan Zhang. On the other hand, Xi-Ru Chen et aldiscussed the strong consistency of the parameter estimation for generalized lin-ear models based on GEE. Base on above research works and the skills used inthe first two parts, this dissertation discusses the strong consistency and the con-vergence rate of MLE for regression coe?cient in reproductive dispersion linearmodels which another type of RDM. In a special case, when the latent roots ofthe design matrix ni=1 XiXiτ/n have the positive lower boundary, MLE leads tothe convergence rate which is the same as the rate what the law of the iteratedlogarithm determined. The conclusion indicates that the convergence rate mainlyrests with the magnitude of the smallest eigenvalue for information matrix, whichare similar to the result of the least square estimation for linear models.