高维模型误差项分布的研究及应用

In the era of “big data”, high dimensional and ultrahigh dimensional data attract much interest, represented by fields like bioinfomatics, statistical genetics, and finance, among others. The increasing data dimension brings tremendous challenges to traditional statistical tools, especially from the perspective of statistical inferences, such as hypothesis testing, confidence interval, prediction, and causal inference. Most statistical inferences, furthermore, rely on the study of the distribution of the error in the model. In this project, we will first discuss the limitations of traditional statistical methods for exploring error distributions based on various high dimensional models. The limitations consist of, but are not limited to, the nonignorable estimation bias when the model dimension is much larger than the sample size. We then develop new estimation methods based on the idea of refitted cross validation, and estimate the error variance function, or density function of error. We will carefully study their asymptotic properties, and provide a range of application scenarios, such as high dimensional causal inferences. There are three main research topics in this project. First, we explore the estimation of error variance function in nonparametric and semi-parametric models, such as varying coefficient models. We will utilize the idea of blocked coordinate descent algorithm and refitted cross validation in the procedure. Secondly, we focus on the modified kernel density estimation of error distribution under high dimensional setting, and prove its asymptotic unbiasedness, consistency and asymptotic normality. We also discuss the bandwidth selection, which is the most essential issue in the kernel density estimation. Finally, we will illustrate the usage of the aforementioned tools in various methodologies. For instance, density estimation of error distribution can be used in high dimensional propensity score matching in causal inference.

在“大数据”时代，高维数据引起人们越来越多的关注。但数据维度的提升为传统分析方法带来很多挑战，特别是统计推断方面，如假设检验、预测、因果推断等。而多数统计推断都需要研究模型误差项分布。我们以此为出发点，讨论在各种模型中，传统方法在误差分布研究中的弊端，如不可忽略的渐进估计偏差等。同时，我们提出基于重复交叉验证思想的误差方差函数估计、误差密度函数估计等，并把改进后的误差项分布估计应用于高维因果推断等领域。我们拟进行三个子项目。第一个子项目中，我们探讨非参数和半参数模型的误差项分布。我们将给出基于分组坐标下降和重复交叉验证思想的方差函数估计方法并研究其渐进理论。第二个子项目集中讨论误差密度函数的估计，给出该估计的相合性、渐进无偏性等，并讨论在核密度估计中至关重要的窗宽选择问题。第三个子项目将给出前两个子项目可以应用的方法论领域，包括稳健回归，分位数回归，及高维因果推断中的倾向评分匹配问题。

本项目聚焦在高维数据的建模中误差项分布问题研究，该研究在假设检验、预测、因果推断等统计推断领域有着广泛的应用。我们以此为出发点，讨论在各种模型中，传统方法在误差分布研究中的弊端，如不可忽略的渐进估计偏差等。为了解决这类问题，我们提出了基于重复交叉验证思想的误差方差函数估计、误差密度函数估计等，并把改进后的误差项分布估计应用于高维因果推断等领域。具体而言，我们进行了三个子项目：（1）我们探讨非参数和半参数模型的误差项分布。我们给出了基于分组坐标下降和重复交叉验证思想的方差函数估计方法并研究其渐进理论。（2）我们集中讨论了误差密度函数的估计，给出该估计的相合性、渐进无偏性等，并讨论了在核密度估计中至关重要的窗宽选择问题。（3）我们给出前两个子项目可以应用的方法论领域，包括稳健回归，分位数回归，及高维因果推断中的倾向评分匹配问题。

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