Research On Two Kinds Of Statistical Models Based On Complex Data

Partly linear model is a kind of semi-parametric model with many practical ap-plications. In1986, Engle first advanced this model, then, there are a lot of papers to study this model. Generalized linear models is used to analyze various types of data. Its application, especially in statistical analysis for the biological、medical、economic and social data, has great significance. Moreover, some complicated data such as missing data, measurement error data and dependent data are often encountered in practice. Statistical analysis for complex data has become one of the frontier and hot issues. Hence, it has theoretical and practical significance to study the statistical inference for partly linear model and generalized linear model with complicated data.The empirical likelihood as a nonparametric statistical method for constructing confidence regions was introduced by Owen in1988. It has many advantages over classic normal approximation based method and the bootstrap method for constructing confidence regions, for example, it does not impose prior constraints on the shape of the region, it does not require the construction of a pivotal quantity and the region is range preserving and transformation respecting. Important features of the empirical likelihood method are its automatic determination of the shape and orientation of the confidence region by the data. Empirical likelihood and its associated properties have been well studied for different statistical models. This thesis applies the empirical likelihood to partly linear models and generalized linear models with complicated data, which extends the application areas of the empirical likelihood method.In this dissertation, we mainly consider the statistical inferences for the partly lin-ear model and generalized linear model with complicated data such as dependent data, measurement error data and missing data. Firstly, we discuss the partly linear model under dependent sequences, and obtain the asymptotic properties of M estimators of parametric and nonparametric components, which extends and improves some known results. Secondly, we study the partly linear model with errors in all variables, and consider the empirical likelihood confidence region of unknown parameters. Lastly, we apply the empirical likelihood method to the generalized linear models, including the empirical likelihood inference for unknown parameter with fixed and adaptive designs, the quasi-likelihood estimation and empirical likelihood with missing data. There are five chapters in this thesis, and the main contents are as follows.In the first chapter, we first introduce some research status and background about the statistical inference for partly linear model and generalized linear model with complex data. Then we recall the related knowledge and theory of the above two statistical models, and discuss the complicated data and main research methods used in this thesis. At last, we list the main work.In the second chapter, the robust M-estimators for the partly linear model under stochastic adapted errors are considered. By use of the piecewise polynomial method and robust estimation method, we obtain the M-estimator of parameter and piecewise polynomial M-estimator of nonparametric function. It is shown that the M-estimator of parameter is asymptotically normal and the M-estimator of the nonparametric func-tion achieves the optimal rate of convergence for nonparametric regression. Some sim-ulations and a real data example are conducted to illustrate the robust properties of the proposed method.In the third chapter, we consider the application of the empirical likelihood method to a partly linear model with measurement errors in possibly all the vari-ables. We use the deconvolution method to deal with the measurement error in non-parametric part and apply the parametric correction for attenuation for overcoming the influence of measurement error in parametric part. It is shown that the empir-ical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Also, a class of estimators for the parameter are constructed, and the asymptotic distributions of the proposed estimators are obtained. Some simulations and an application are given to illustrate the proposed methods.In the fourth chapter, empirical likelihood for generalized linear model with fixed and adaptive designs is investigated. When the responses are multidimensional and design matrices are fixed or adaptive, it is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter and the resulting estimator is shown to be asymptotically normal. Some simulations are used for illustration.In the fifth chapter, we discuss the quasi-likelihood estimation and empirical like-lihood for generalized linear model with miss data. Firstly, based on quasi-likelihood equation, we obtain the quasi-likelihood estimator of the unknown parameter and prove the strong consistency and the convergence rate of the proposed estimator. Then we use the empirical likelihood method to construct the empirical likelihood confidence region of parameter with missing responses ar random, and simulation studies are given to interpret the proposed method under three missing mechanisms.