高维统计模型中的稳健推断及其应用

In this project, we will study robust inference in three kinds of statistical models(ultra-high dimensional linear model, generalized partially varying coefficient single-index model, generalized additive partial linear model). For ultra-high dimensional linear model, we will study a robust variable screening method based on weighted Wilcoxon type estimation. Based on U-process theory, we will study the sure independence screening property. Furthermore, the proposed method can be used to deal with ultra-high dimensional partial linear model. The estimation efficiency of our method will be demonstrated through extensive comparisons with other methods by simulation studies and one real data example. For generalized partially varying coefficient single-index model, a class of B-spline based sieve robust quasi-likelihood estimation will be proposed and the asymptotic properties of the proposed estimators will be discussed. Besides, we will study the problem of providing a test statistic, more resistant to outliers than classical methods,to decide between a linear and a semiparametric model. Simulation studies will be conducted to examine the small-sample properties of the proposed estimates and a real dataset will be used to illustrate our approach. For generalized additive partial linear models, we will employ B-spline smoothing for estimation of nonparametric functions, and derive robust quasi-likelihood based estimators for the linear parameters. We will further develop a class of variable selection procedures for both the linear parameters and the nonparametric component functions by employing a nonconcave penalized robust quasi-likelihood, which will be shown to have an asymptotic oracle property. Monte Carlo simulations and an empirical example will be presented for illustration.

本项目主要研究近年来被广泛关注且具有重要应用前景的三类统计模型(超高维线性模型，广义变系数单指标模型,广义可加部分线性模型)中的稳健方法。对于超高维线性模型，我们将研究基于加权Wilcoxon型估计的稳健变量扫描方法。利用U过程理论，我们将建立该方法的sure screening 性质，并利用此方法来研究超高维部分线性模型的稳健变量扫描问题。我们将通过模拟和实际数据分析来验证所提方法的优良性。对于广义变系数单指标模型，将研究基于样条方法的稳健拟似然估计，并通过构造稳健拟似然比统计量来研究假设检验，通过模拟和实际数据分析来说明所提出的方法的可行性。对于广义可加部分线性模型，本项目将利用稳健拟似然和B 样条方法，研究参数分量与非参数分量的稳健估计，获得估计的大样本性质，利用惩罚似然方法对参数分量与非参数分量进行稳健变量选择，并将所提出的方法应用于实际数据分析。

本项目主要研究近年来被广泛关注且具有重要应用前景的几类统计模型(超高维可加危险率模型和高维线性模型，非线性常微分方程模型,区间删失数据半参数模型)中的稳健统计推断。对超高维可加危险率模型，我们考虑了右删失数据下模型中低维系数的假设检验问题，提出一种称为 variance reduced partial profiling estimator (VRPPE)的方法，该方法使我们能够在维数远大于样本量时能够检验每一个回归系数的显著性; 在协变量维数与样本量比值趋近于一个大于0小于1的常数条件下，利用广义F检验研究了线性模型回归系数的假设检验问题； 对非线性常微分方程模型研究当解析解不容易获得时非线性常微分方程模型中常系数、变系数的稳健估计和基于数值离散算法和局部线性估计方法的两阶段估计方法;研究了在信息删失 (informative censoring) 假设下， I 型区间删失数据比例风险回归模型.我们建议一种基于 I 样条的半参数极大似然估计方法，并利用 Copula 模型来描述失效时间和删失变量之间的依赖关系，所提出的方法适用于研究边际回归对相依假设的敏感性分析.