Several Studies On Linear Models And Single Index Models

Robust estimation and variable selection are two important aspects in the statistical modeling. The goal of variable selection is to find those covariates which are truly related to the response variable. So it can reduce the complexity of model and improve the prediction accuracy. Meanwhile we hope that the variable selection method is robust, especially when many outliers exist. Robust variable selection methods should can resist the effect of outliers and perform stably. On the other hand, longitudinal data has a broad application in the biological medicine, economics, sociology and many other fields. In recent years, it becomes one of most active topics in the statistical science. This paper mainly studies robust estimator, variable selection and longitudinal data analysis for linear models, genalized linear models, single index models and single index coefficient models.In Chapter 2, based on the SCAD penalty function and rank regression, a robust variable selection approach is proposed for linear regression models with a diverging number of parameters. The proposed method is resistant to heavy tailed errors or outliers in the response. The proposed estimator enjoys the consistency and oracle property under some regularity conditions. In order to solve the difficulties in calculation of existing methods, we propose a very fast and efficient greedy coordinate descent algorithm to compute the penalized rank regression. In order to deal with p(29)n, we propose a two-step estimation by using the sure independence screening procedure based on the distance correlation. We further prove that the proposed two-step estimation enjoys the oracle property. Finally, extensive simulations are carried out to verify the robustness and efficiency of the proposed approach.In Chapter 3, since the linear model of Chapter 2 can not handle discrete response variables, we will study robust estimator and variable selection for generalized linear models with longitudinal data. Specifically, we construct new robust and efficient estimation equations based on the exponential score function and weight function. The proposed method can achieve robustness against outliers both in the response and the covariate domain. In order to avoid solving the convex optimization problem,robust and efficient smooth-threshold generalized estimating equations are constructed to obtain the estimators of parameters and select the significant variables simultaneously. Under some regularity conditions, we prove that the proposed estimators possess the consistency and oracle property. Furthermore, we prove that the proposed estimation is robust through the influence function. Finally, simulations and real data analysis are given to assess the finite sample performance.In Chapter 4, the problem of estimation for single index models with longitudinal data is considered. Firstly, we obtain initial estimators of the index coefficient and nonparametric link function by ignoring the possibile correlation between repeated measures. Secondly, in order to avoid estimating correlation coefficient matrix in the generalized estimating equations, we use the modified Cholesky decomposition to decompose the covariance matrix as autoregressive coefficients and innovation variances, which can be estimated by utilizing regression modeling. Thirdly, we employ profile weighted least squares technique to construct more efficient two-step estimators for the index coefficients and nonparametric link function. Under some regularity conditions, we prove that the proposed estimators possess the consistency and asymptotic normality. Finally, simulation study and real data analysis are used to confirm the superiority of the proposed approach.In Chapter 5, for the single index coefficient model, a robust and efficient estimation procedure is proposed based on the local linear technique and modal regression. Under some regularity conditions, we prove that the proposed estimator enjoys the consistency and asymptotic normality. We further discuss the optimal bandwidth in theory and give the method of choosing bandwidth in practice. We prove that the proposed method does not lose the efficiency of estimator. Finally, simulation studies are carried out to evaluate the robustness and efficiency of the proposed approach.In Chapter 6, the problem of estimation for single index coefficient models with longitudinal data is considered. Since the estimation of nonparametric link function in Chapter 5 involves undersmoothing bandwidth, which brings challenge for the selection of bandwidth in practice. Thus, this chapter presents centralized genalized estimating equations to overcome such drawback. To improve the efficiency of statistical inference, we adopt the modified Cholesky decomposition approach to obtain the estimator of the covariane matrix. So we can construct more efficient centralized generalized estimating equations for the index coefficient and then employ weighted least squares technique to obtain more efficient estimator of nonparametric link function. Under some regularity conditions, we establish large sample properties of the proposed estimators. Finally, simulation studies and real data analysis are conducted to verify the efficiency and practicability of the proposed approach.