Robust Estimation In Semiparametric Mixed Models

Some assumptions are essential when people make statistical inference according to the data sets. However, these assumptions can be hardly consistent with the actual situation and are just its approximate description. It is usually expected that the small difference between the real data sets and the statistical models used to describe them have little influence on the final conclusions. But it is not always true. Recent several decades, it has been found that the deviation of the statistical models from the real data sets will seriously affect most classical statistical methods. Therefore, more attention begin to be paid on the robust statistical methods. The so-called "robust statistical methods" refer to those being insensitive to the deviation of statistical models from the real data sets. In other words, the small difference between the model assumptions and the real data sets will have little influence on these robust methods. In the middle of 1980's, Green et al. (1985) and Engle et al. (1986) independently proposed a kind of important statistical models, i.e. semiparametric models when they respectively studied the agriculture experiment and the relation between weather and electricity sales. Based on the semiparametric models, semiparametric mixed models are developed, which include fixed and random effects, and parametric and nonparametric components. These semiparametric mixed models combine many advantages of parametric models, nonparametric models and mixed models. They can fully utilize the information of the data and be more close to the actual situation. Furthermore, generalized semiparametric mixed models are the natural extension of the semiparametric mixed models and generalized linear models.The robust estimation in semiparametric mixed models are discussed in this thesis, including:1. In the first chapter, we give a brief description of the semiparametric mixed models. The backgrounds and present development of robust statistics are introduced. In addition, we also address the backgrounds and development of generalized estimating equations methods. Finally, the main results of this thesis are introduced.2. In the second chapter, we mainly study the robust estimation of mean components in generalized semiparametric mixed models , including robust estimation of regression parameters and nonparametric function. First, we construct robust estimating equations with conditional expectation based on approximation of nonparametric part by B-spline ; Second, we estimate the conditional expectation involved in the robust estimating equa- tion by drawing samples of random effects from their posterior distribution through Monte Carlo Markov Chain (MCMC) algorithm; Third, we show that the asymptotic properties of the proposed robust estimators under some regular conditions; Fourth, simulations are carried out to investigate the performance of the resulting estimators and compare their efficiency with those proposed by He, Fung & Zhu (2005) in the normal semipaxametric mixed models. And we find our robust estimators have higher efficiency in the case of outliers. Finally, to illustrate that our method is available, four real data sets are analyzed by the proposed robust methods.3. In the third chapter, we mainly study the robust estimation of covariance parameters in semiparametric model with continuous response. First, a set of robust estimating equations for estimation of both mean and covariance parameters are constructed. Second, the asymptotic properties of the proposed robust estimators are shown under some regular conditions. Third, some simulations are made to see the performance of the resulting robust estimators. Finally, two real data sets are analyzed by the proposed robust method.4. In the fourth chapter, we research the robust estimation of both mean and variance components in the generalized semiparametric mixed models based on the idea of "robustified likelihood function". First, we adopt the penalized spline method to approximate the nonparametric function. Second, we give the robust estimators of mean and variance components resulted from the "robustified likelihood function". Third, robust MCMC method also resulted from the robustified likelihood function is proposed. Fourth, the asymptotic properties of the robust estimators are presented. Fifth, simulations are carried out to see the performance of the resulting robust estimators. Finally, two real data sets are analyzed by the proposed robust method.5. In the fifth chapter, we study the robust estimators of correlation parameters and the corresponding bias-corrected robust estimators in generalized linear models. First, we introduce the method of bias correction. Second, we propose three robust estimating equations about the correlation parameters and give the corresponding bias-corrected estimators. Third, some simulations are made to see the performance of the proposed robust and bias-corrected robust estimators. Finally, two real data sets are analyzed by the proposed robust methods.In summary, the robust estimation of mean and variance components in semiparametric mixed models are systematically studied in this thesis. For the estimation of mean components, we propose the robust estimators of mean components by constructing ro- bust estimating equations and extend the results of He, Fung & Zhu (2005) and Sinha (2004); For the estimation of variance components, we first study the estimation of covariance parameters in semiparametric models, then we propose the robust estimator of variance components in generalized semiparametric mixed models and propose the robust MCMC method based on the idea of "robustified likelihood function", finally we consider the robust estimators of correlation parameters and the corresponding bias-corrected robust estimators. We extend and develop the work of Huggins (1993), McCulloch(1997), Mills, Field & Dupuis (2002), Yau & Kuk (2002) and Sinha (2006). The effectivity of the proposed robust estimators is justified by the rich simulation studies. These results are not only useful in theory but also significant in practice.