Robust Inference And Model Selection Methods For Some Semiparametric Models

The semiparametric model not only retains the good interpretability of parameter model and the flexibility of nonparametric model, more importantly, it can effectively circumvent the problem of “dimensionality curse”(Bellman, 1961) that encountered by the nonparametric model. So this model has attracted much interest and attention of statisticians. Although there have been lots of related literature on semiparametric model, most of them are committed to the mean regression approach, based on least squares, profile least square, likelihood function and profile likelihood function method. It is well known that the mean regression approach probably loss much efficiency under nonnormal error distributions despite it has been proved to be the most efficient one in the case of normal error. Therefore, it is necessary to establish some robust as well as efficient estimation method. In addition, model selection plays a vital role in statistical modeling for the purpose of improving the simplicity and predictive accuracy of the model, so how to select an optimal model from a set of candidate models, and then carry out some statistical inference is particularly necessary. As a result, the main attention of this thesis is devoted to researching some robust estimation and model selection methods of several different semiparametric models. Specifically, the following chapters are included.Chapter 2 of this thesis considers the estimation and variable selection of single index model based on local walsh-average regression. We firstly obtain the estimators of single index parameter as well as nonparametric function via a walsh-average-based iterative procedure. Under some suitable assumptions, the large sample properties of the resulting estimators are established. Then based on the theoretical results, we further derive the asymptotic relative efficiencies of the proposed method versus the least squares as well as least absolute regression approaches. In addition, a penalized estimation of single index parameter is considered via a combination of SCAD penalty and local walsh-average regression to simultaneously do estimation and variable selection, and a modified BIC criterion are proposed to select the penalty parameter. The oracle property of variable selection is also established. Finally, numerical simulations and a real data analysis are conducted to verify the robustness and efficiency of the proposed method.Chapter 3 of this thesis studies the robust least absolute regression method of partially linear single index model. We firstly estimate the single index parameter via an iterative algorithm, and a two-step estimation procedure is followed to obtain the estimators of linear parameter and nonparametric function. In addition, the penalized estimation of the linear part in this model is considered, by combining the adaptive Lasso penalty and least absolute regression, so as to do estimation and variable selection, simultaneously. Large sample properties of the obtained estimators and the oracle property of variable selection are established under some suitable assumptions. Finally, numerical simulations and a real data analysis are conducted to confirm the excellent performance of the proposed method.Chapter 4 of this thesis proposes a robust and efficient estimation method of partially linear single index model considered in Chapter 3, by integrating the ideas of local modal regression and step by step concept. Large sample properties of the obtained estimators of parametric and nonparametric parts are established under some suitable assumptions. In order to obtain a sparse estimator of the linear part of this model, the SCAD penalty and the method of local modal regression are combined to simultaneously do estimation and variable selection, and the oracle property of variable selection are also established. Moreover, a revised expectation maximization algorithm is presented to solve the issue of computation. Finally, numerical simulations and a real data analysis are conducted to verify the robustness and efficiency of the proposed method.Chapter 5 of this thesis studies the asymptotic properties of two different estimation methods of partially linear single index model. Xia et al.(2002) proposed the outer product of gradients estimation procedure based on least squares regression, but did not prove the large sample property in theory. Therefore, we first theoretically prove the asymptotic normality of the estimators resulted by Xia et al.(2002). Taking into account of the fact that the efficiency of least squares method is susceptible to the influence of outliers and thick tail distribution, a robust and efficient estimation method based on local rank regression and outer product of gradient approach is proposed to estimate the single index parameter in this model, and the corresponding asymptotic properties are established under some suitable assumptions. Furthermore, based on the theoretical results of the studied two methods, asymptotic relative efficiencies of the corresponding parametric and nonparametric portion are derived. Finally, numerical simulations and a real data analysis are carried out to compare the performance of the two methods.Chapter 6 of this thesis considers the robust estimation and model identification of semiparametric additive model. Firstly, we propose a robust model identification technique of this model based on B-spline approximation, local modal regression and a double SCAD penalty. Under some suitable assumptions, we establish the theoretical properties of the proposed method, which can correctly identify the zero components, linear components and nonlinear components from the original model, and the resulting estimate of linear parameter has oracle property. Besides, a revised expectation maximization algorithm is presented to solve the issue of computation. Finally, the results of numerical simulation further validate the excellent performance of the proposed method.