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迴歸分析中，若變數間有非線性(nonlinear)的關係，此時我們可以用B-spline線性迴歸，一種無母數的方法，建立模型。Penalized spline是B-spline方法的一種改良，其想法是增加一懲罰項，避免估計函數時出現過度配適的問題。本文中，考慮三種方法:(a) Marginal Mixed Model approach, (b) Conditional Mixed Model approach, (c) 貝氏方法建立信賴帶，其中，我們對第一二種方法內的估計式作了一點調整，另外，懲罰項中的平滑參數也是我們考慮的問題。我們發現平滑參數確實會影響信賴帶，所以我們使用cross-validation來選取平滑參數。在調整的cross-validation下，Marginal Mixed Model的信賴帶估計不平滑的函數效果較好，Conditional Mixed Model的信賴帶估計平滑函數的效果較好，貝氏的信賴帶估計函數效果較差。 / In regression analysis, we can use B-spline to estimate regression function nonparametrically when the regression function is nonlinear. Penalized splines have been proposed to improve the performance of B-splines by including a penalty term to prevent over-fitting. In this article, we compare confidence bands constructed by three estimation methods: (a) Marginal Mixed Model approach, (b) Conditional Mixed Model approach, and (c) Bayesian approach. We modify the first two methods slightly. In addition, the selection of smoothing parameter of penalization is considered. We found that the smoothing parameter affects confidence bands a lot, so we use cross-validation to choose the smoothing parameter. Finally, based on the restricted cross-validation, Marginal Mixed Model performs better for less smooth regression functions, Conditional Mixed Model performs better for smooth regression functions and Bayesian approach performs badly.

Master of Science / Department of Statistics / Kun Chen / We study the composite group bridge penalized regression methods for conducting bilevel variable selection in high dimensional linear regression models with a diverging number of predictors. The proposed method combines the ideas of bridge regression (Huang et al., 2008a) and group bridge regression (Huang et al., 2009), to achieve variable selection consistency in both individual and group levels simultaneously, i.e., the important groups and the important individual variables within each group can both be correctly identi ed with probability approaching to one as the sample size increases to in nity. The method takes full advantage of the prior grouping information, and the established bi-level oracle properties ensure that the method is immune to possible group misidenti cation. A related adaptive group bridge estimator, which uses adaptive penalization for improving bi-level selection, is also investigated. Simulation studies show that the proposed methods have superior performance in comparison to many existing methods.

Doctor of Philosophy / Department of Statistics / Weixin Yao and Kun Chen / Ordinary least-squares (OLS) estimators for a linear model are very sensitive to unusual values in the design space or outliers among y values. Even one single atypical value may have a large effect on the parameter estimates. In this proposal, we first review and describe some available and popular robust techniques, including some recent developed ones, and compare them in terms of breakdown point and efficiency. In addition, we also use a simulation study and a real data application to compare the performance of existing robust methods under different scenarios. Finite mixture models are widely applied in a variety of random phenomena. However, inference of mixture models is a challenging work when the outliers exist in the data. The traditional maximum likelihood estimator (MLE) is sensitive to outliers. In this proposal, we propose a Robust Mixture via Mean shift penalization (RMM) in mixture models and Robust Mixture Regression via Mean shift penalization (RMRM) in mixture regression, to achieve simultaneous outlier detection and parameter estimation. A mean shift parameter is added to the mixture models, and penalized by a nonconvex penalty function. With this model setting, we develop an iterative thresholding embedded EM algorithm to maximize the penalized objective function. Comparing with other existing robust methods, the proposed methods show outstanding performance in both identifying outliers and estimating the parameters.

One situation in survival analysis is that the failure of an individual can happen because of one of multiple distinct causes. Survival data generated in this scenario are commonly referred to as competing risks data. One of the major tasks, when examining survival data, is to assess the dependence of survival time on explanatory variables. In competing risks, as with ordinary univariate survival data, there may be explanatory variables associated with the risks raised from the different causes being studied. The same variable might have different degrees of influence on the risks due to different causes. Given a set of explanatory variables, it is of interest to identify the subset of variables that are significantly associated with the risk corresponding to each failure cause. In this project, we develop a statistical methodology to achieve this purpose, that is, to perform variable selection in the presence of competing risks survival data. Asymptotic properties of the model and empirical simulation results for evaluation of the model performance are provided. One important feature of our method, which is based on the idea of the L1 penalized Cox model, is the ability to perform variable selection in situations where we have high-dimensional explanatory variables, i.e. the number of explanatory variables is larger than the number of observations. The method was applied on a real dataset originated from the National Institutes of Health funded project "Genes related to hepatocellular carcinoma progression in living donor and deceased donor liver transplant'' to identify genes that might be relevant to tumor progression in hepatitis C virus (HCV) infected patients diagnosed with hepatocellular carcinoma (HCC). The gene expression was measured on Affymetrix GeneChip microarrays. Based on the current available 46 samples, 42 genes show very strong association with tumor progression and deserve to be further investigated for their clinical implications in prognosis of progression on patients diagnosed with HCV and HCC.

In this thesis we present an alternative to quantiles, which is known as expectiles. At first we define the notion of expectile of a distribution of ran- dom variable and then we show some of its basic properties such as linearity or monotonic behavior of τ-th expectile eτ in τ. Let (Y, X), Y ∈ R, X ∈ Rp be a ran- dom vector. We define conditional expectile of Y given X = x, which we denote eτ (Y |X = x). We introduce model of expectile regression eτ (Y |X = x) = x⊤ βτ , where βτ ∈ Rp and we examine asymptotic behavior of estimate of the regression coefficients βτ and ways how to calculate it. Further we introduce semiparametric expectile regression, which generalizes the previous case and adds restrictions on the estimate of the regression coefficients which enforce desired properties such as smoothness of fitted curves. We illustrate the use of theoretical results on me- chanographic data, which describe dependence of power and force of a jump on age of children and adolescents aged between 6 and 18. Keywords: expectiles, expectile regression, quantiles, penalized B-splines 1

Background: Many questions in statistical genomics can be formulated in terms of variable selection of candidate biological factors for modeling a trait or quantity of interest. Often, in these applications, additional covariates describing clinical, demographical or experimental effects must be included a priori as mandatory covariates while allowing the selection of a large number of candidate or optional variables. As genomic studies routinely require mandatory covariates, it is of interest to propose principled methods of variable selection that can incorporate mandatory covariates. Methods: In this article, we propose the ridge-lasso hybrid estimator (ridle), a new penalized regression method that simultaneously estimates coefficients of mandatory covariates while allowing selection for others. The ridle provides a principled approach to mitigate effects of multicollinearity among the mandatory covariates and possible dependency between mandatory and optional variables. We provide detailed empirical and theoretical studies to evaluate our method. In addition, we develop an efficient algorithm for the ridle. Software, based on efficient Fortran code with R-language wrappers, is publicly and freely available at https://sites.google.com/site/zhongyindaye/software. Results: The ridle is useful when mandatory predictors are known to be significant due to prior knowledge or must be kept for additional analysis. Both theoretical and comprehensive simulation studies have shown that the ridle to be advantageous when mandatory covariates are correlated with the irrelevant optional predictors or are highly correlated among themselves. A microarray gene expression analysis of the histologic grades of breast cancer has identified 24 genes, in which 2 genes are selected only by the ridle among current methods and found to be associated with tumor grade. Conclusions: In this article, we proposed the ridle as a principled sparse regression method for the selection of optional variables while incorporating mandatory ones. Results suggest that the ridle is advantageous when mandatory covariates are correlated with the irrelevant optional predictors or are highly correlated among themselves.

A seleção de variáveis é uma importante fase para a construção de um modelo parcimonioso. Entretanto, as técnicas mais populares de seleção de variáveis, como, por exemplo, a seleção do melhor subconjunto de variáveis e o método stepwise, ignoram erros estocásticos inerentes à fase de seleção das variáveis. Neste trabalho, foram estudados procedimentos alternativos aos métodos mais populares para o modelo de riscos proporcionais de Cox e o modelo de Cox com fragilidade gama. Os métodos alternativos são baseados em verossimilhançaa penalizada e diferem dos métodos usuais de seleção de variáveis, pois têm como objetivo excluir do modelo variáveis não significantes estimando seus coeficientes como zero. O estimador resultante possui propriedades desejáveis com escolhas apropriadas de funções de penalidade e do parâmetro de suavização. A avaliação desses métodos foi realizada por meio de simulação e uma aplicação a um conjunto de dados reais foi considerada. / Variable selection is an important step when setting a parsimonious model. However, the most popular variable selection techniques, such as the best subset variable selection and the stepwise method, do not take into account inherent stochastic errors in the variable selection step. This work presents a study of alternative procedures to more popular methods for the Cox proportional hazards model and the frailty model. The alternative methods are based on penalized likelihood and differ from the usual variable selection methods, since their objective is to exclude from the model non significant variables, estimating their coefficient as zero. The resulting estimator has nice properties with appropriate choices of penalty functions and the tuning parameter. The assessment of these methods was studied through simulations, and an application to a real data set was considered.