复杂函数型数据的同时置信带统计推断

In recent years, functional data have emerged in various fields and have attracted researchers’ much attention. Due to their complex dependence structures, functional data present great challenges to statisticians. How to construct a valid estimator for unknown functions in modeling such data in order to make useful statistical inferences is an important problem to be solved in this research project. It is well known that a simultaneous confidence band is a type of area estimate obtained from the observed data that contains the true curve of an unknown function with a predetermined error probability. It is a powerful tool for making inferences on the global shape of a true function. In this project we propose to construct asymptotic simultaneous confidence bands for different parameters of interest of different types of functional data, specifically, (i) constructing oracally efficient spline estimator with asymptotically correct simultaneous confidence band for the population mean function of sparse functional data with spikes; (ii) constructing oracally efficient two-step spline-kernel estimator with asymptotically correct simultaneous confidence band for variance function of dense functional data; (iii) constructing oracally efficient estimators with asymptotically correct simultaneous confidence band for the link function of partially linear single-index models for sparse functional data.

近年来函数型数据在各个领域不断涌现而备受研究者们的关注。其复杂的相依结构，对统计学相关研究提出了巨大挑战。如何对此类数据模型下的未知函数构造有效估计并进行合理统计推断是本项目拟解决的重要问题。众所周知，同时置信带是在未知函数的定义域上构造的一个区域，使得该区域抓住整条未知函数的概率等于特定的置信水平。它是对未知曲线进行整体统计推断的有力工具。本项目期望对不同类型的函数型数据模型的未知参数（未知函数）进行构造同时置信带研究，具体包含：(i) 对带有跳点的稀疏性函数型数据的总体均值函数构造默示有效样条函数估计和渐近精确的同时置信带; (ii) 对稠密性函数型数据的方差函数构造两步默示有效样条-核函数估计和渐近精确的同时置信带; (iii) 对稀疏性函数型数据的局部线性单指标模型的链接函数构造默示有效估计和渐近精确的同时置信带。

在当今数字时代，复杂数据分析技术全面渗透到各行各业。本项目对复杂函数型数据以及观测不完全数据建立了一系列统计推断方法。具体包含：（1）稀疏型纵向数据的部分线性单指标模型的整体统计推断方法建立；（2）协变量随机缺失下非参数回归曲线差异的同时置信带构造；（3）协变量随机缺失下非参数回归均值函数曲线的同时置信带构造；（4）固定设计自变量非参数回归下均值函数之差的同时置信带构造；（5）协变量随机缺失下方差函数曲线的同时置信带构造；（6）扩展部分线性单指标模型的整体统计推断方法建立。