基于oracle有效性的高维面板与函数型数据统计推断

Along with the rapid development of information science and internet technology, increasingly more panel data and functional data come into being, such as, stock return rate series in the financial market, electrocardiograms, and quantitative near infrared reflectance spectroscopy data. However, being with high-dimensional, dependent and latent information, high-dimensional panel and functional data have brought challenges to data analysis. There are several problems with the existing methods. First, since the explanatory variables are unobservable, least-squares or quantile regression fail in data analysis. Second, the efficiency of taking residuals as errors has not been verified by statistical theory, which leads to the failure of further statistical inference, such as the construction of Simultaneous Confidence Region (SCR). Third, it is difficult to make use of trajectories, which are realizations of a random function in functional data.. Based on oracally efficient estimation of high dimensional data and functional data, this project concerns three statistical inference issues. First, based on the estimated factor returns and residual errors, we will study the factor return and specific error distribution for large panel data and propose the oracally efficient estimators of such distributions under moderate assumptions, which can be used to construct SCR, a desirable tool for the global inference. Next, we are going to propose oracally efficient estimators of the corresponding error distribution in functional data, expecting to obtain the asymptotic normality of the estimators and derive the theoretical framework of SCR construction. Last, we would like to apply the statistical inference of latent information from the above two complex data models to real data analysis. . Solutions to the above problems not only enrich the theory of high-dimensional panel and functional data in the background of statistical inference, support the important application of the statistical theory and provide new ways to conduct the research on high dimensional data analysis and prediction, but also highlight the practical guidance in areas such as option pricing, risk management, weather forecast, ecological analysis and so on.

随着信息技术的发展，生产生活中广泛出现面板和函数型数据，如股票日收益、心电图、光谱数据等。但高维面板与函数型数据高维、相依、具有潜变量信息的特征，给数据分析带来挑战。现有统计方法存在如下困难：解释变量不可观测，最小二乘或分位数回归失效；残差代替误差的有效性，理论性质缺乏，无法做进一步的统计推断，如同时置信区域的构造；样本轨道信息的重建。本项目将紧密围绕高维面板和函数型数据相关估计的oracle有效性展开研究。具体涉及三个研究板块：第一、高维因子模型因子和误差分布函数的oracale有效估计及同时置信区域构造方法；第二、函数型数据误差分布函数的oracle有效估计及同时置信区域构造方法；第三，相应的统计分析算法与应用研究。以上研究将丰富高维面板和函数型数据的统计推断理论，为高维数据分析和预测等问题的研究提供新思路，在期权定价、风险管理、气象预报、生态分析等领域也具有现实指导意义。

宏观经济数据、脑/心电图、生物测定数据等复杂数据都包含作为隐性信息的测量误差，尽管通过样本方差、协方差、均值等统计量也能获知数据的部分信息，但面对日益增长的复杂数据，由于其分布中蕴藏着更为丰富的业务知识和决策信息，如何充分有效地进行分布函数的估计并从中提取有价值的可用信息仍是一个极具挑战性的问题。从统计学的角度来看，分布函数、协方差函数和潜在变量信息的获知，对于解决分布检验、分位数估计、模型预测、风险价值等问题具有重要启发意义。基于此，本项目致力于解决：(1)高维面板数据、时间序列数据分布函数的统计推断及应用；(2)函数型数据误差分布函数、协方差函数的统计推断、正定性分析及应用；(3)面板半参数分位数回归模型深度学习框架的构建及应用。本文的研究重点是以非参数统计方法为基本支撑，基于复杂数据背景，搭建集oracle有效估计理论、非参数光滑化方法、SCR方法、深度学习框架、利用数据驱动原理选择光滑化参数进行模拟与实际数据分析等功能的平台。本项目解决的关键科学问题是：潜变量的预处理；非参数光滑化方法；统计推断理论；oracle有效性的证明；SCR构造方法；复杂数据下半参数模型分位数回归的深度学习框架。本项目的创新点是：1) 理论上的特色和创新：本项目以估计量的oracle表现为着眼点，呈现集复杂数据误差分布函数的oracle有效估计与SCR方法于一体的完整框架，为复杂数据分析的理论和应用研究提供基础依据和指导意见。2) 应用上的特色和创新：本项目提出的带有潜变量的复杂数据光滑化技术、误差分布函数的信息重建，能够适用于复杂数据的多领域交叉研究，实现从统计理论到应用的跨越，为应用科学研究提供坚实的统计学基础。以上研究将丰富高维面板和函数型数据的统计推断理论，为高维数据分析和预测等问题的研究提供新思路，在期权定价、风险管理、气象预报、生态分析等领域也具有现实指导意义。