QUANTILE REGRESSION FOR ANALYZING CORRELATED DATA IN ULTRA-HIGH DIMENSION

Data plays a crucial role in driving statistical development. Statistics is dedicated to developing effective methods for data mining,  information extraction, and theoretical support. However, the rapid progress in science and technology intensifies the complexity of data characteristics, including ultrahigh dimensionality, strong correlations, non-normality, outliers, and incomplete data. These characteristics bring new challenges and opportunities to the development of statistics and data science. Notably, the prevalence of intercorrelation in high-dimensional data significantly impacts the stability of traditional variable selection methods. Consequently, high-dimensional statistical inference methods relying on penalized estimation may fail. To address these challenges, robust statistical methods are urgently required. This thesis focuses on the ultrahigh-dimensional statistical analysis and investigates the issues of feature screening, variable selection, and hypothesis testing within quantile regression models. By exploring the theoretical foundations and practical methods of quantile regression models in ultrahigh-dimensional data analysis, this thesis aim to provide new strategies and insights for handling complex data.

Feature screening is a valuable approach for reducing data dimensionality. To reduce ultrahigh-dimensional correlated features to a manageable dimension, this thesis introduces a novel approach called the Quantile Ridge Regression (QRR) screener. Ridge regression, known for its collinearity-friendly penalized estimation, serves as a bridge to feature screening. By establishing the Bahadur representation of the QRR estimator, this thesis demonstrates that the QRR screener possesses the sure screening property, ensuring that it can correctly screen all important variables with a probability tending to one in ultrahigh-dimensional settings. To address the computational challenges posed by the non-smoothness of the quantile loss function, this thesis applies the accelerated and generalized Alternating Direction Method of Multipliers (ADMM) to efficiently solve the ultrahigh-dimensional quantile ridge regression problem. Numerical simulation results underscore the superiority of the QRR screener over existing marginal or conditional measure-based screening methods, exhibiting a higher accuracy screening rate for a given model size. Moreover, the proposed screening method is not limited to traditional linear regression problems but can also be applied to survival analysis. By leveraging the inverse probability weighting method, the censored QRR screener outperforms existing methods in handling right censored data.

Variable selection is essential for achieving unbiased estimation. Building upon the successful application of the QRR screener, this thesis further proposes a two-stage variable selection method, namely QRR-LIPS, relied on the likelihood-based discrepancy measure. This method accurately and efficiently identifies important variables through implementing nested signal detection on ranked features. However, inactive variables strongly correlated with important variables can easily lead to a placeholder effect, interfering with the accuracy of variable selection. To address this issue, this thesis innovatively introduces a greedy algorithm based on forward search and competition mechanism to enhance the vanilla QRR-LIPS method. This improvement endows the variable selection process with an error-correction mechanism, effectively preventing overfitting and underfitting in extreme cases. Theoretically, this thesis proves the strong selection consistency of both the vanilla and enhanced versions of the QRR-LIPS. Numerical simulation results show that, the two-stage QRR-LIPS methods exhibit significant computational efficiency and accurate variable selection ability compared to penalized methods based on either the full feature space or the reduced space after screening. Furthermore, the enhanced QRR-LIPS performs more robustly under dependent design and can effectively cope with complex correlation structures in data. Notably, the proposed method extends beyond ordinary linear regression, offering applicability to semiparametric and nonparametric regression models. By leveraging the B-spline approximation method, the proposed two-stage methods can effectively address the high-dimensional varying coefficient models and high-dimensional additive models with correlated data, outperforming existing methods in finite samples.

Hypothesis testing is the theoretical guarantee of statistical inference. To establish rigorous and reliable testing procedures, this thesis proposes a Wald-type testing framework for linear structures of parameters of interest within high-dimensional regression models. In the estimation process, this thesis employs a convolution-smoothed quantile regression framework to effectively solve the non-smoothness issue of the empirical loss function. To ensure valid inference, this thesis adopts the idea of sample splitting, performs model selection on one subset, parameter estimation on another subset, and utilizes the cross-fitting method to combine statistics from these subsets. However, without Oracle property, the presence of spurious and redundant parameters often leads to biased estimation in the covariance matrix, affecting the power of the test. To address this challenge, this thesis proposes an information-weighted fused estimator. This novel method adequately combines the information from two subsamples through matrix weighting, thereby reducing estimation bias. Theoretically, this thesis proves the asymptotic normality of the fused estimator and derives the limiting distribution of the Wald-type test statistic based on this fused estimator under the null hypothesis and local alternative hypothesis. Empirical results show that the proposed test statistic can control the Type I error and achieve high testing power under a series of linear hypotheses.

Data analysis serves as the ultimate orientation of statistical research. To demonstrate the practical applicability of the proposed methods, this thesis applies the series of proposed quantile regression methods for ultrahigh-dimensional correlated data to the feature screening, variable selection, and targeted inference analysis of ultrahigh-dimensional anticancer drug sensitivity genes.The outcomes of our analysis align with existing biomedical literature, providing valuable insights for in-depth understanding of drug action mechanisms and the optimization of clinical treatment strategies.

数据是统计学科发展的源动力。统计学致力于开发挖掘数据、提炼信息的有效方法，并为之提供扎实的理论支持。随着科技的迅猛发展，各个领域的数据呈现出了诸多复杂特性，如超高维、强相关、非正态分布、离群值存在、数据不完整等现象。这些特性为统计与数据科学的发展带来了新的机遇与挑战。高维数据中普遍存在的相关性，使得传统变量选择方法的稳定性受到了严重影响，进而可能导致广泛依赖于惩罚估计的高维统计推断方法失效。面对这些不太理想的数据，亟需更加稳健的统计方法予以支撑。本文主要针对超高维数据的背景，研究分位数回归模型的特征筛选、变量选择、以及假设检验等问题。通过深入探讨分位数回归模型在超高维数据分析中的理论基础和实践方法，旨在为处理复杂数据提供新的思路和策略。

特征筛选是缩减数据的可靠途径。为了将超高维相关性数据降低到可操作的维度，本文提出了分位数岭回归(Quantile Ridge Regression, 记为QRR)筛选器。岭回归是对共线性友好的惩罚估计，架起通往特征筛选的桥梁。通过建立QRR估计的Bahadur表达式，本文证明了QRR筛选器具有确定筛选性质(Sure Screening Property)。这一性质保证了在超高维情形下，该筛选器能够以趋于一的概率正确筛选所有重要变量。为了解决分位损失函数的不可微带来的计算挑战，本文应用广义加速的交替方向乘子法(Alternating Direction Method of Multipliers, 记为ADMM)来高效地求解超高维分位数岭回归问题。数值模拟结果表明，与现有的边际或条件度量的筛选方法相比，QRR筛选器展现出了明显的优越性。其在给定模型大小下具有较高的准确筛选率。本文所提出的筛选方法不仅仅局限于传统的线性回归问题，还能够被巧妙地应用于生存分析领域。通过利用逆概率加权的方法，所提出的筛选方法能够有效地处理删失数据，并在现有方法中展现出优势。

变量选择是准确估计的必由之路。在成功应用QRR筛选器后，本文进一步提出了一种基于似然差异度量的变量选择方法(Likelihood-Based Post-Screening Selection Algorithm, 记为LIPS)，与QRR配套形成两阶段变量的QRR-LIPS方法。该方法通过对排序的特征实现嵌套式地信号检测，能够准确、快速地分隔出重要变量。然而，与重要变量具有强相关性的虚假变量容易产生占位效应，干扰变量选择的准确性。为了解决这一问题，本文创新性地引入了前向搜索和竞争机制相结合的贪婪算法，对QRR-LIPS方法进行了改进。这一改进赋予了变量选择过程纠错机制，防止了极端情况下的欠拟合和过拟合现象的发生。理论上，本文证明了基础版和改进版QRR-LIPS方法的变量选择强相合性。数值模拟结果表明，与基于全特征空间或筛选后降维空间的惩罚方法相比，两阶段QRR-LIPS方法展现出明显的计算效率和准确的选择变量的能力。此外，改进版的QRR-LIPS方法在相依型设计矩阵下的表现更稳健，能够有效应对数据的复杂相关结构。本文所提出的两阶段QRR-LIPS方法不仅仅局限于简单的线性回归问题，还能扩展到半参数和非参数回归模型中。通过利用B样条近似的方法，所提出的变量选择方法能够有效解决相关数据下高维变系数模型和高维可加模型的变量选择问题，并在有限样本下优于现有方法。

假设检验是统计推断的理论保障。为了构建严谨可靠的检验方法，本文针对高维回归模型中感兴趣参数的线性结构，提出了Wald型检验方法。在估计过程中，本文采用了卷积平滑的分位数回归框架，巧妙地解决经验损失函数的不可微问题。为了确保有效的推断，本文借鉴了样本分割的思想，在一个子集上进行模型选择，在另一个子集上进行参数估计，并利用交叉拟合方法来结合统计量。然而，在缺乏Oracle性质的估计下，冗余参数的存在往往造成协方差矩阵的估计偏差，进而影响检验统计量的功效。为了应对这一挑战，本文提出了一种基于信息加权的融合估计方法。该方法通过矩阵加权方法充分结合了两个子样本的信息，有效地减少了估计偏差。理论上，本文证明了融合估计的渐近正态性，推导了建立在该估计上的Wald型检验统计量在零假设和局部备择假设下的渐近分布。数值模拟结果表明，所提出的检验统计量在一系列线性假设下能够控制第一类错误和并且具有较高的检验功效。

数据分析是统计研究的最终导向。为了展示所提出方法的实际应用价值，本文将提出的一系列面向超高维相关数据的分位数回归方法，运用到对超高维抗癌药物敏感性基因的选择和目标基因的推断分析中。分析结果与现有的生物医学文献相吻合，为深入理解药物作用机制和优化临床治疗方案提供了宝贵的见解。

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