Some Studies On The Semiparametric Generalized Linear Model

When we bring data set into one regression model, some assumptions are used, for example, homogeneity of variance, et al. Even the model itself is an assumption. If the assumptions aren't true, the estimate and other statistical inference will be influenced. So the people begin to study the diagnostics and the testing of homogeneity of variance for regression models, which they are important procedures on dealing with regression problems and have important signification in theory and practice. In the middle of 1980's, Green, et al(1985), while studying agriculture experiments, and Engle, et al(1986), while studying the relation between weather and electricity sales, proposed independently an important statistical model, that is semiparametric regression model. There are the parametric part and nonparametric part in the model. A lot of actural problems can be describled through this model. The informations of data can be used fully and it is more true than parametric models and nonparametric models. The semiparametric generalized linear models are the natural extension of the semiparametric regression models and generalized linear models. In this paper, the statistical diagnostics and the variance component testing are studied for these models. The main results of this paper are as follows:1. In the first chapter we give a brief description of semiparametric regression models, the Cross Validation and the Generalized Cross Validation of choosing the smoothing parameter. Generalized Cross Validation is used in this paper. The background and present conditions for statistical diagnostics are introduced. Secondly, We also introduce the background and present conditions for ordinary regression models, generalized regression models and longitudinal data models. At the last, the score test statistics are introduced. The score test statistics for variance component testing and outlier are used in this paper.2. In the second chapter, we study systematically the statistical diagnostics and influence analysis for semiparametric generalized linear models. The main results are as follows: Under the estimate of smoothing spline, we first show that the case deletion model is equivalent to mean shift outlier model and the diagnostic statistics such as Cook dis-tance,et al are introduced.The score test statistics for outlier tests are derived and a series of simple counting formulas for diagnostic statistics are obtained. Then the local influence measures are discussed and the counting formulas of influence matrices for case weightsperturbation model, mean shift perturbation model and arguments perturbation model are obtained. Finally, two numerical examples(the data of mice medicine experiments and the data of oldman urinary incontinent)are used to illustrate that our method is available.3. In the third chapter, we study systematically the statistical diagnostics and influence analysis for semiparametric generalized linear mixed model. Firstly, the equivalency of case deletion model and mean shift outlier model is investigated and some diagnostic statistics such as residuals, leverage and Cook distance,et al are obtained. The score test statistics for outlier tests are also studied. Secondly, The local influence for the models is investigated and the counting formulas of influence matrices for case weights perturbation model and mean shift perturbation model are obtained. At last, a real example (the data of children respiratory infection)is provided to illustrate that our method is available.4. In the fourth chapter, the testing of variance component in semiparametric generalized linear model is studied. For discrete random effect model, the testing of variance component is studied and the score test statistics are obtained. For continuous model, we study it's variance component testing for nonconstant dispersion parameter, random regression coefficients and two situations respectively and the score test statistics are obtained. The methodology is seen to perform better for small and moderate sample sizes...