Statistical Analysis Of Nonlinear Reproductive Dispersion Mixed Models

There are lots of clustered data or correlated data in the fields of biomedicine, statistical genetics, engineering, economics, educational psychology, sociology, etc.. Random effects models are powerful tools for analyzing these data. If clustered data or correlated data are approximately normally distributed and can be described by linear structure linear mixed models are suitable tools for analyzing them. However, with the rapid development of science and technology, the accuracy in data analysis is highly expected and the clustered data or correlated data that are non-normally distributed or cannot be described by the linear structure become more and more. Therefore the theory and method of random effects models have been developed and extended. To meet the need of the above development, random effects models have been extended from original linear mixed models to nonlinear mixed models, generalized linear mixed models, nonlinear exponential family mixed models and nonlinear reproductive dispersion mixed models. Since the above models are the special case of nonlinear reproductive dispersion mixed models that are widely applied, the study of nonlinear reproductive dispersion mixed models plays an important role in the theory of random effects models and even in the whole regression theory.The dissertation completely study statistical analysis for nonlinear reproductive dispersion mixed models. The main content of this dissertation is as follows: 1. We study maximum likelihood estimations (MLE) of parameters for nonlinear reproductive dispersion mixed models. Due to nonlinearity, non-normal random errors and random effects, marginal likelihood function of observed data involves complex and intractable high-dimensionality integral. In general, the analytical form of the integral is not available. Thus, we treat random effects as hypothetical missing data and apply M-H algorithm to obtain the EM algorithm and the stochastic approximation algorithm of MLE respectively. We also use Laplace approximation approach to investigate nonlinear reproductive dispersion mixed models. After marginal log-likelihood function is approximated with Laplace approximation method to avoid direct calculation of high-dimensionality, Fisher Score iterative method is used to obtain MLE based on the approximate log-likelihood function. Finally, we use simulation studies and real examples to illustrate the above methods and conduct comparison analysis for these methods.2. We make Bayesian analysis for nonlinear reproductive dispersion mixed models. Via treating random effects as hypothetical missing data and combining Gibbs sampling and M-H algorithm, a hybrid algorithm is used to produce the joint Bayesian estimates of parameters and random effects.3. The large sample properties for the MLE of nonlinear reproductive dispersion mixed models are investigated. After applying Laplace approximation method to approximate marginal log-likelihood function of observed data and proposing some regular conditions, the MLE is proved to be existent, strong consistent and asymptotically normally distributed.4. We propose a geometric framework and study confidence regions of para- meters and subset parameters fbr nonlinear reproductive dispersion mixed models. Via applying differentiable geometric method, we obtain a geometric frame in Euclid inner product space based on the Laplace approximate marginal likelihood. Three improved approximate confidence regions of parameters and subset parameters are investigated for nonlinear productive dispersion mixed models based on the proposed curvatures.In summary, this dissertation firstly investigate estimations for nonlinear reproductive dispersion mixed models, then the large sample properties for the MLE for these models, finally confidence regions for parameters and parameter subsets in terms of curvatures from geometric point of view. The results are useful base for investigating further nonlinear reproductive dispersion models.